Quasi-isometric recognition of virtually polycyclic groups

Proof. We give the decomposition argument to specify precisely which parts of the Cartan action may be nonsemisimple; see also [3, Section 4.D] for the Cartan grading. For v,u\in\mathfrak d, nilpotence of \mathfrak d gives

for sufficiently large b. Consequently \ad u preserves every generalized eigenspace of \ad v. Indeed, between two primary blocks with distinct eigenvalues, the commutator with \ad v is invertible, so an operator killed by a power of that commutator has no such block. All eigenvalues are real by real triangulability. Successively taking the primary decompositions for a basis of \mathfrak d therefore gives simultaneous invariant primary spaces. On each such space all diagonal entries of a triangular representation agree on that basis, and hence agree as linear forms on \mathfrak d. Their common form \gamma is a character, since diagonals of commutators vanish. This proves the direct sum decomposition into the spaces in (2.1).

The derivation identity shows that weights add under brackets: applying \ad v-(\gamma(v)+\eta(v))\id repeatedly to [x,y], with x\in\mathfrak g_\gamma and y\in\mathfrak g_\eta, gives a binomial sum involving powers of \ad v-\gamma(v)\id on x and of \ad v-\eta(v)\id on y. Every summand vanishes at a sufficiently large power. Nilpotence of \mathfrak d gives \mathfrak d\subseteq\mathfrak g_0. If this inclusion were strict, the induced action of \mathfrak d on \mathfrak g_0/\mathfrak d would consist of nilpotent operators. Engel’s theorem would give a nonzero fixed class. Any representative of that class normalizes \mathfrak d, contrary to self-normalization. Thus \mathfrak g_0=\mathfrak d.

The nonzero weight spaces are \mathfrak d-invariant. Their generated subalgebra \mathfrak n is therefore \mathfrak d-invariant, and \mathfrak g=\mathfrak n+\mathfrak d makes it an ideal. If \gamma\ne0, choose v\in\mathfrak d with \gamma(v)\ne0. The restriction of \ad v to \mathfrak g_\gamma is invertible, so \mathfrak g_\gamma\subseteq[\mathfrak g,\mathfrak g]. In a simultaneous real triangularization of \ad(\mathfrak g), every element of \ad([\mathfrak g,\mathfrak g]) is strictly upper triangular. Engel’s theorem applied to the derived algebra proves its nilpotence, and hence that of \mathfrak n.

The images of the generating nonzero weight spaces span \mathfrak n/[\mathfrak n,\mathfrak n], and its induced weight decomposition has no zero summand. Choose a homogeneous basis of this quotient and homogeneous lifts X_i. These lifts generate \mathfrak n: if \mathfrak h is their generated subalgebra and \mathfrak n^j denotes the lower central series, then \mathfrak n=\mathfrak h+\mathfrak n^2. Induction, using brackets, gives \mathfrak n=\mathfrak h+\mathfrak n^j for every j, and nilpotence concludes the argument.

We now pass from Lie algebras to groups. Dixmier’s real-root criterion implies that \exp_G:\mathfrak g\to G is an analytic diffeomorphism [6, Theorem 3]. The inclusion of either nilpotent subalgebra integrates from its simply connected Baker–Campbell–Hausdorff group, and this homomorphism is the restriction of \exp_G in exponential coordinates. It is injective, with closed image, because \exp_G is a diffeomorphism and each subalgebra is a closed vector subspace. Injectivity of \exp_G also gives the stated description of N\cap D. This proves the claims about N,D, including normality of N.

The group law in the domain of q is

which verifies the homomorphism assertion. Its derivative at the identity is (x,v)\mapsto x+v, which is onto. Its image is therefore an open subgroup of the connected group G, so it is all of G; translation also makes q a submersion everywhere. The kernel formula follows directly from nd=1. Local smooth sections of a submersion, restricted to relatively compact neighborhoods, give compact sets of lifts; a finite cover proves the claim for an arbitrary compact set.

If \mathfrak n=0, then \mathfrak g=\mathfrak d is nilpotent. Conversely, in a nilpotent \mathfrak g all adjoint operators are nilpotent. The decomposition just proved then has only weight zero, so \mathfrak g=\mathfrak d and \mathfrak n=0.□

Proof. We first bound the coordinates of a short path. For the reverse estimate, we construct uniformly bounded products of degree-one exponentials in the rescaled groups. The remaining assertions follow from these coordinates and the Haar Jacobian.

A path of length at most t splits its endpoint into a product of at most C(1+t) increments from a fixed compact neighborhood. Their logarithms lie in a fixed bounded set. In a product of a exponentials, the degree-b part of its formal logarithm has a sum of absolute coefficients at most C_b(1+a)^b. To see the bound, expand the product of formal exponentials through degree s in the noncommuting variables, and then use the finite expansion of \log(1+u) in those degrees. The degree-b coefficient sum of the product is at most a^b/b!; the finitely many compositions of b appearing in the logarithm give the asserted bound. Applying the degree-b Dynkin projection, which sends an associative monomial to its iterated bracket and acts by multiplication by b on homogeneous Lie polynomials, converts this expansion to brackets with only a degree-dependent coefficient bound. A bracket involving more than j increments belongs to \mathfrak d^{j+1} and has zero V_j projection. Evaluating the remaining finitely many types of brackets on the bounded logarithms therefore bounds this projection by C_j(1+a)^j. This proves (2.6).

For the converse, we first record the limit of the transported laws. The inclusion [V_i,V_j]\subseteq\bigoplus_{k\ge i+j}V_k shows that the coefficient of a bracket from V_i\times V_j into V_k, when transported by \delta_r, is multiplied by r^{i+j-k}. For r\ge1 these coefficients are bounded and converge to the associated graded bracket. The finite Baker–Campbell–Hausdorff formula proves convergence of the group laws and all their derivatives on compact sets. Denote the transported group by D_r; its underlying exponential coordinate space is the fixed vector space \bigoplus_jV_j.

We now produce uniformly bounded products of degree-one exponentials covering a fixed coordinate neighborhood. Choose a basis e_1,\ldots,e_a of V_1. These vectors generate the graded Lie algebra: by the definition of the lower central series, the quotient \mathfrak d^j/\mathfrak d^{j+1} is spanned by brackets of length j of classes in \mathfrak d/\mathfrak d^2. In D_\infty, let H consist of all finite products of \exp_\infty(t e_i) and put

This space contains the e_i and is invariant under every \Ad_{\exp_\infty(t e_i)}. Differentiation makes it invariant under \ad e_i, so it contains all their brackets and equals the entire graded Lie algebra. Choose b=\dim D such conjugated vectors \Ad_{h_\ell}e_{i_\ell} forming a basis, where each h_\ell is a fixed finite product of degree-one exponentials. The map

is the identity at t=0 and has these basis vectors as its derivative columns there.

Use the same fixed words for h_\ell in D_r to define \Phi_r. The maps \Phi_r converge with their derivatives to \Phi_\infty on compact parameter sets. The inverse function theorem therefore supplies a fixed parameter ball and a fixed coordinate neighborhood U of the identity contained in their images for all sufficiently large r. More explicitly, choose a parameter ball on which the derivative of \Phi_\infty differs by less than one quarter of the smallest singular value of D\Phi_\infty(0) from that derivative; the corresponding estimate with one half holds for \Phi_r. The usual contraction proof of the inverse function theorem then gives one common image ball. Every element of U is thus a product of a fixed number of \exp_r(c e_i), with all |c| bounded by one constant. The conjugators here are themselves fixed words in these exponentials.

A fixed number of powers of U covers the coordinate box \{\sum_jv_j:\|v_j\|\le1\}, uniformly in r: choose an integer M large enough that every vector v in this box satisfies v/M\in U, and use \exp_r(v)=(\exp_r(v/M))^M. Under \Delta_r, each degree-one factor becomes \exp_D(rc e_i), whose intrinsic length is at most r|c|\|e_i\|_D, where the last norm is the Riemannian norm at the identity. The resulting products have length at most Cr. Enlarging C handles the remaining compact interval of r\ge1. Given arbitrary v_j, apply this bound with r=\max(1,\max_j\|v_j\|^{1/j}) to obtain (2.7).

For the equicontinuity assertion, the logarithm of the difference of z_r and z'_r in the transported law is a vector \xi_r=\sum_j\xi_{r,j} tending to zero. The logarithm of the corresponding difference in D is \delta_r\xi_r. Thus (2.7) gives

The compact length bound follows from the same coordinate estimate. The converse compactness assertion follows from (2.6) after dividing the V_j coordinate by r^j. Equation (2.9) follows directly from convergence of the transported product, since \delta_r(tv)=rtv on V_1.

Finally, the differential of the exponential map, after left translation to the identity, is a polynomial in \ad v with constant term \id. Since \ad v is nilpotent, this polynomial has determinant one. This proves the claims about Haar measure on both nilpotent groups. The linear coordinate map \delta_r has determinant r^Q. The coordinate bounds (2.6) place a radius-t ball in a product of Euclidean balls with radii C(1+t)^j, proving the upper volume bound. Conversely, (2.7) puts the coordinate box \{\sum_jv_j:\|v_j\|\le(t/(2C))^j\} inside the radius-t ball when t\ge2C. This box has volume a fixed positive constant times t^Q. Decreasing that constant handles 1\le t\le2C, and proves (2.10). These two bounds give doubling for t\ge1. At sufficiently small radii a smooth coordinate chart and the Riemannian metric give volume comparable to t^{\dim D}; the remaining compact interval of radii is handled by monotonicity and positivity of ball volume. This proves (2.11). The proof used only simple connectivity and nilpotence, so it applies without change to the group L in the last assertion.□

Proof. The restrictions of the diagonal characters \chi_j to \mathfrak d are exactly the generalized weights of its action on \mathfrak g, with their multiplicities. All \chi_j vanish on \mathfrak n by Lemma 2.1. In particular every Cartan weight vanishes on \mathfrak d\cap\mathfrak n, and, being a character, also on [\mathfrak d,\mathfrak d].

Homogeneous bracket words in the X_i span \mathfrak n. Their weights are sums of the \alpha_i, whereas every \alpha_i itself occurs. The weights on \mathfrak d=\mathfrak g_0 are zero. It follows that

Since \mathfrak n\subseteq\mathfrak k and \mathfrak g=\mathfrak n+\mathfrak d, inclusion gives \mathfrak g/\mathfrak k\cong\mathfrak d/(\mathfrak k\cap\mathfrak d), and passing through V proves (2.12). It also shows that every weight of \mathfrak n descends to W. The statements about the integrated maps follow from simple connectivity. If \nu=0, Lemma 2.1 gives nilpotence and W=0. Conversely, if W=0, all adjoint characters vanish, so every adjoint operator is strictly triangular and Engel’s theorem makes G nilpotent.

For d=\exp_D(v) and b_\gamma=\dim\mathfrak n_\gamma, write

The map v\mapsto T_\gamma(v) is linear, and exponentiation gives

Replacing v by -v gives its inverse. By (2.6), \|v\| is bounded by a polynomial in 1+|d|_D. The two finite sums therefore prove (2.13), uniformly over the finitely many weight spaces, including the zero weight when it occurs.

For v\in\mathfrak d, the action of \ad v on \mathfrak g/\mathfrak n\cong \mathfrak d/(\mathfrak d\cap\mathfrak n) is nilpotent. Unimodularity of G consequently gives

Thus \det(\Ad_d|_{\mathfrak n})=1. Bracket addition in (2.1) proves that S_w preserves brackets. Choosing v\in\mathfrak d over w in (2.16) also gives

An automorphism of a simply connected nilpotent group changes Haar measure by its Lie algebra determinant, so both asserted actions preserve Haar measure. As N and D are unimodular and the action preserves Haar measure on N, direct substitution in the semidirect product law proves that dn\,dd is left Haar measure on N\rtimes D.

Finally, \pi_* annihilates \mathfrak d^2, so in the splitting (2.4) it depends only on V_1. The same projection is a homomorphism on the associated graded group. Equation (2.15) now follows from \delta_r|_{V_1}=r\id.□

Proof. Write a word Y\in\mathcal Z in a maximizing basis as Y=\sum_{Z\in\mathcal B}c_{Y,Z}Z. Whenever c_{Y,Z}\ne0, replacing Z by Y gives another basis. Maximality gives h_Y(E)\le h_Z(E). Thus

has uniformly bounded coefficients. There are only finitely many unscaled bases and words, so their coefficients admit a common bound. The reverse inclusion follows because the basis words belong to the list. This proves the first assertion.

The bracket of two basis words is either zero or a word of the list, with exponent equal to the sum of their exponents. Formula (3.5) therefore bounds every normalized structure constant. The Baker–Campbell–Hausdorff formula terminates at degree m, which proves the assertion for a fixed number of factors. To control k factors, expand the formal product of their exponentials and then its logarithm, retaining only degrees at most m. In degree j there are at most C_m(1+k)^j choices of factor indices, and each term contains at most j input coordinates. Converting these finitely many homogeneous Lie polynomials into brackets costs constants depending only on m. The asserted polynomial bound in k,L follows from the bounded normalized structure constants.

Haar measure of a simply connected nilpotent group in exponential coordinates is a fixed multiple of Lebesgue measure. Indeed the Jacobian factor of the exponential map is \det((1-e^{-\ad v})/\ad v)=1, interpreted by its finite power series, since \ad v is nilpotent. The determinant of the scaled basis is its unscaled determinant times e^{rP(E)}. The finitely many unscaled determinants are bounded above and away from zero. This proves (3.3).

For an exact weight dilation, h_Z(E+w)=h_Z(E)+\gamma_Z(w). Every basis has total weight zero: it is a weight-homogeneous basis of \mathfrak n, and the total weight is the trace character, which vanishes by Lemma 2.3. Hence maximizing bases remain maximizing after a translation of E, and the claimed shift follows.

For conjugation, first remove the scalar action on each weight space. Lemma 2.3 bounds the remaining operator and its inverse polynomially in 1+|d|_D. Words of weight \alpha_i span the weight space containing X_i, and each such word satisfies

Consequently the residual action on a scaled letter is a polynomially bounded combination of scaled words. Applying brackets and using the already established structure-constant bounds proves the same assertion for every scaled word. Restoring the character action is the exact shift by \pi d/r. This gives (3.4); replacing d by d^{-1} proves its inverse version.□

Proof. Every basis from \mathcal Z contains all the letters X_i: all longer words vanish in \mathfrak n/[\mathfrak n,\mathfrak n], and the letters project to its specified basis. As observed above, the sum of the word weights of any such basis is zero. Expanding this identity in letters gives

The total exponent of a basis is unchanged by translating E, which proves translation invariance. The other two assertions about P follow from the corresponding properties of support functions and the finiteness of the list.

For a segment [0,w], each support is \max(0,\alpha_i(w)). These expressions give positive homogeneity, piecewise linearity, and the triangle inequality for p, the last by taking the maximum of the basis sums after applying subadditivity term by term. Translation invariance gives p(-w)=p(w). If w\ne0, the weights span W^*, and (3.8) implies that some \alpha_i(w)>0. All word exponents on the segment are nonnegative, and the mandatory letter X_i makes every basis sum positive. Thus p is a norm.

Every member of \mathcal A is a linear isometry of this norm. Its unit ball is a full-dimensional bounded polytope, so its linear isometry group injects into the permutation group of its finitely many vertices. In particular \mathcal A is finite.

Suppose A\in\mathcal A and A^*\alpha_j is not on one of the original oriented rays. From a Euclidean ball K centered at zero, remove a sufficiently small cap at the support point of A^*\alpha_j. The resulting compact convex set E has the same supports in all original directions \alpha_i, but a strictly smaller support in direction A^*\alpha_j. Its supports in every other direction can only decrease. Since each candidate basis sum for P(AE) contains the letter j, all these finitely many sums decrease strictly. Thus P(AE)<P(AK)=P(K)=P(E), contrary to A\in\mathcal A. Pullback therefore maps the finite set of rays into itself; applying the same argument to A^{-1} proves that it permutes them.□

Proof. We construct a submersive scheme for each limiting normalized group. Its persistence under nearby group laws will then give uniform generation by compactness.

Pass first to a subsequence with the same maximizing basis. The normalized structure constants are bounded by Lemma 3.1, so a further subsequence converges. The Jacobi identity and the nilpotence-class bound pass to the limit, and its group law is the limiting BCH polynomial. Every letter is a basis vector. More generally, the bracket word giving a basis vector satisfies the exact identity

Thus its normalized value is the same coordinate vector throughout the sequence and in its limit. The limiting letters therefore still bracket-generate the whole limiting algebra.

Here is a finite endpoint scheme with full rank in this limiting group. Let H be the subgroup generated by its letter one-parameter subgroups, and consider

This space contains the letters and is invariant under \Ad_{\exp(tX_i)}. Differentiating proves its invariance under \ad X_i, so it contains all brackets and is the entire algebra. Choose d_N vectors \Ad_{h_j}X_{i_j} forming a basis, with each h_j a finite product of generator exponentials. The map

sends zero to the identity and has these basis vectors as its differential columns at zero. Expanding the h_j and their inverses gives a generator scheme as in (3.9). At a parameter point having fixed entries for the conjugating words and zero entries for the inserted increments, its differential is surjective. Enlarge a fixed parameter cube so this point is interior.

Use the same words and fixed parameters for the converging laws. Their endpoint maps converge with derivatives on compact parameter sets. The inverse function theorem, applied to the indicated d_N variables with the others fixed, supplies one fixed identity neighborhood contained in their images for all sufficiently late indices. Its local inverse supplies continuous parameter choices. For a fixed-radius log cube choose k so large that division of every log vector by k lies in this neighborhood. The identity \exp(v)=(\exp(v/k))^k covers the cube by a fixed number of copies of the scheme.

This proves the uniform generation assertion by contradiction: any sequence violating every proposed common length and parameter bound has a convergent normalized-law subsequence to which the preceding argument applies. Equivalently, the compact closure of the normalized laws has a finite cover by neighborhoods carrying the preceding inverse charts. There are only finitely many maximizing bases. One may pad words by identities, and concatenate the finitely many schemes, so there is no measurable choice among infinitely many alphabets. Local inverse choices on finite Borel partitions give the stated measurable endpoint choices.

For the growing-radius assertion let a_0=\min_i\|\alpha_i\|>0. Enlarge E to E+\varepsilon_r\overline B_W(1), where \varepsilon_r=o(1) is large enough that r\varepsilon_ra_0 dominates the logarithm of the radius and the fixed box-comparison constants. Every word exponent increases by at least \varepsilon_ra_0. The all-word comparison then embeds the original growing box in a fixed-radius box for the enlarged template. Generation there changes each letter amplitude by only e^{o(r)}.

Finally choose one sufficiently large centered ball as template. All its word exponents are strictly positive, so its boxes exhaust N as r\to\infty. The uniform number of factors just established proves the last assertion.□

Proof. If \|\log_Nn\|\le e^r, choose a fixed large centered ball E whose every word exponent is at least one. The finiteness of the unscaled bases gives n\in B_r(E,L_0) for a fixed L_0. Lemma 3.3 writes it as a bounded product of letter factors with amplitudes at most Me^{rs_i(E)}. Choose v_i\in\mathfrak d with \alpha_i(\pi v_i)=1. Conjugating a letter factor back by d_i=\exp_D((rs_i(E)+b\log(2+r))v_i) removes its exponential amplitude. For a sufficiently large fixed b, the extra logarithmic term also dominates the polynomial unipotent factor from Lemma 2.3. The conjugated-back log vector is bounded, while |d_i|_G=O(r+1). The resulting outward, bounded, and return moves prove the first estimate, also for bounded n.

By left invariance the distance for a switch is the length of \Ad_{d^{-1}}\exp(tX_i). Its logarithm has norm at most

The first estimate makes this o(r) in the stated regime and O(\log(2+r)) in the exact regime.□

Proof. Choose, for each letter, a support point v_i\in E_V for \alpha_i\circ\pi. Fix a linear section V\to\mathfrak d. From the pivot visit d_i=d_0\exp_D(r\widetilde{v_i-v_0}), along the corresponding one-parameter segment. Its abelianization traces the segment r[v_0,v_i]\subset rE_V, and its length is O(r). Decompose u using Lemma 3.3. To apply its next letter factor, move along this leg on the current D-chart, switch at d_i, and reverse the leg on the new sheet. The support equality and Lemma 3.4 give the claimed switch cost. After the final return the endpoint is ud_0.

Every continuous homomorphism l kills N, since its differential kills \mathfrak n\subset[\mathfrak g,\mathfrak g], and factors on D through V. It therefore has the specified values on the legs. It is Lipschitz for the fixed metrics, so each short switch changes its values by at most its Lipschitz constant times the switch length. This proves the third assertion.

Measurable generator choices are supplied by Lemma 3.3. Short connecting paths may also be chosen measurably. The Riemannian exponential map on a closed tangent ball of the asserted length radius is continuous from a compact space and covers the relevant metric ball. Such a map has a Borel section: use nested finite closed covers of successively smaller diameters and choose the first subpiece whose image contains the endpoint; all these image sets are compact, and the nested choices have one limiting point. The corresponding geodesics give the required choices. The same argument gives the subexponential version.

For the specialization to W, choose an affine section of \pi:V\to W through a_D(d_0)/r over \pi d_0/r, with one fixed linear part, and apply the construction to the image of E. All leg displacements depend only on differences in that affine section. Their bounds therefore do not depend on its translating kernel component; the bound on |d_0|_D/r controls the positions. For a prescribed curve, replace the displayed legs by its portions between the pivot and a support position, and reverse each portion after the switch. All estimates remain the same.□

Proof. Translate the starting point to the identity and subdivide the path into at most C(1+r) increments of bounded length. Lemma 2.1 supplies bounded lifts (v_j,e_j) of these increments. Their accumulated product has coordinates

Thus |d_j|_D\le C(1+r), and \pi d_j is the actual height of the corresponding path vertex.

Expand \log_Nv_k in the fixed weight spaces. The polynomial residual action gives, in a weight \gamma, coefficients bounded by C(1+r)^b e^{\gamma(\pi d_{k-1})}. Words of that weight span its space, and \max_E\gamma\le h_Z(E) for every such word Z. Consequently each conjugated increment has normalized coordinates bounded polynomially in r. The product estimate in Lemma 3.1 gives the asserted bound for u_j.

Bounded right thickening replaces (u,d) by (u\Ad_dv,de) with (v,e) bounded. The same estimate at the terminal height controls this extra factor and proves the thickening assertion.

Equip N\rtimes D with a fixed left invariant Riemannian metric. The homomorphism q is globally Lipschitz: its derivative bound at the identity is transported everywhere by left translations. Therefore a-separated projected endpoints have separated chosen lifts, with separation bounded below by a fixed positive number. Small fixed-radius balls around those lifts are disjoint. The thickening assertion places all of them in

Product Haar is valid by Lemma 2.3; the first factor has volume at most a polynomial times e^{rP(E)}, and the second has polynomial volume by Lemma 2.2. Every small lifted ball has the same positive Haar volume. This proves (3.11). Injectivity of q is not used.

For the sequential statement, enlarge E by a vanishing-radius ball so it contains the actual relative path heights divided by r and zero. The support functional changes by o(1), and the box exponents by o(1). Polynomial factors are subexponential. For an unrestricted path of length t, its intermediate heights are O(1+t) because \pi is Lipschitz. The same lifted-increment argument now bounds each coordinate by e^{O(1+t)}, and its polynomially many BCH terms preserve this bound.□

Proof. If N=1, then G=D is nilpotent. The endpoints lie in the radius-C_0t ball, and disjoint fixed-radius balls about separated points, together with Lemma 2.2, give the polynomial bound directly. Suppose now that N\ne1. Repeat the bounded-increment lifting in the preceding proof. Every accumulated D-coordinate has intrinsic length O(1+t) and height norm at most M. On each of the finitely many weight spaces the character factor is bounded by a constant depending on M, while the residual action is polynomial in 1+t. Hence every conjugated N-increment has polynomial log norm. There are O(1+t) increments; the fixed-degree nilpotent product formula bounds the log norm of their product by C(1+t)^b. The same bounds survive fixed bounded right thickening. The allowed N log-coordinate region and the intrinsic D-ball both have polynomial Haar volume. Disjoint small balls around lifts of separated endpoints give the result exactly as before. In particular this is a finite polynomial bound; it does not follow by interpreting a subexponential error as polynomial.□

Proof. Set K_a=N\cap\overline B_G(1,a). This is compact since N is closed and the metric on G is proper. If nd and n_0d have distance at most a, then n_0^{-1}n\in\Ad_dK_a. Conjugation by d preserves N-Haar, so the possible n have measure at most \vol_N K_a, independently of d. A maximal a-separated subset of the bounded slice covers it by radius-a balls. Summing the preceding measure bound gives the assertion, after a harmless increase of a or of C_a at boundary points. Left translation by x is an isometry. In the product maximum metric the covering balls are products, so the same proof multiplies the constants.□

Proof. If the set is nonempty, fix one representative (u_0,d_0). Every other projected point differs from it by an element of a fixed radius-2R ball. Choose a bounded lift (v,e) of that difference, using Lemma 2.1. In G,

It follows that

The first factor has log norm at most C(1+T)^b e^{cH} by the character and unipotent bounds. The second has polynomial log norm by intrinsic nilpotent coordinates in D. The logarithm of a in D and in N is the same vector in \mathfrak d\cap\mathfrak n, because both are restrictions of the injective exponential map of G. The finite BCH formula therefore places u_0^{-1}u in an N log-coordinate ball of radius C(1+T)^{b'}e^{c'H}. Its Lebesgue, hence Haar, volume has the stated form after changing the constants. This proves the bound even for outer measure and uses no centrality of the overlap.□

Proof. We first keep a definite part of E away from the zero set of the chosen derivative minor. A coordinate slice then retains positive measure, and a small cube in that slice makes the restricted map injective. Change of variables on this cube will give the image-volume bound.

The first step uses a scalar sublevel estimate. If a polynomial Q in k\ge1 variables has degree at most d\ge1 and some coefficient of absolute value at least c, then, for R\ge1 and 0<\eta<c,

No upper coefficient bound is needed here.

For k=1, let the sublevel set have length l>0. It contains d+1 points whose mutual distances are at least l/[2(d+1)]: after fewer than d+1 choices, deleting intervals of that radius about the chosen points removes total length strictly less than l. Lagrange interpolation at these points bounds every coefficient of Q by

Indeed the denominators are products of d point differences, while the coefficients of the numerator polynomials are bounded by C_d(1+R)^d. The asserted lower bound on one coefficient now proves (4.3) in one variable.

For the induction step, write

and choose q_j which contains a coefficient of absolute value at least c. Put u=\eta/c\in(0,1) and \tau=c u^{(k-1)/k}. The induction hypothesis bounds the (k-1)-dimensional measure of \{|q_j|\le\tau\} in the base cube by

Above its complement, the one-variable estimate bounds the length of each sublevel slice by

Fubini’s theorem, using the bound 2R on the remaining slice lengths, proves (4.3). The argument also applies to constant polynomials, since d is only a degree upper bound.

Return to the minor Q of D\Phi in the statement. Its degree is at most b(D-1), so we can use d=\max\{1,b(D-1)\}. Choose a compact K\subset E with |K|\ge m/2. For a sufficiently large constant C_0, depending only on a,b,D, set

Since m\le(2R)^a, we have 0<\eta<c_0\le1 after increasing C_0. Estimate (4.3), with coefficient lower bound c_0, makes the bad-minor set have measure at most m/4. Therefore the compact set

has measure at least m/4.

Permute the domain coordinates so that the chosen minor uses the first b columns, and write x=(v,z) with v\in\R^b. Fubini supplies a fixed z for which the compact slice K'_z satisfies

When a=b, there is no slicing and we use K' itself. For f(v)=\Phi(v,z), its derivative and the Lipschitz constant of its derivative on [-R,R]^b are both at most

We now retain a small cube on which f is injective. Cover [-R,R]^b by at most C_b((1+R)/\ell)^b closed cubes of side at most

One such cube J contains at least the reciprocal of this number times the mass in (4.4). Choose v_0\in K'_z\cap J. The least singular value of Df(v_0) is at least \eta/L^{b-1}, since its determinant has absolute value at least \eta. For every v\in J,

Integrating along a line segment in J gives

Thus f is injective on J. Change of variables on its interior, with the boundary omitted as a null set, gives

The image here is compact and is contained in \Phi(E). Combining the cube count and (4.4) yields

Substituting \eta and L proves (4.1) with the stated exponents. This proof uses neither finite global fibers nor a global multiplicity bound.

Finally, a polynomial map submersive somewhere has a derivative minor which is a nonzero polynomial. Coefficient convergence gives a fixed positive lower bound for one coefficient of that same minor of \Phi_r, for all sufficiently large r, and a fixed upper bound for all coefficients of \Phi_r. Every factor in (4.1) then has logarithm bounded below by -o(r) under the hypotheses on R_r and |E_r|. The final variant follows from the same displayed estimate.□

Proof. For a Borel set A\subset Q, let

This is a Borel function. Right Haar invariance, the substitution z=x s_1, and Fubini give

For the first inequality, the inner function is supported in Q\supp\sigma_j, has integral \mu(A), and Cauchy–Schwarz applies. Compactness makes the containing set measurable and of finite measure.

Starting with S_h=S, work backwards and define

The contribution from S_j\setminus S_{j-1} to the preceding integral is at most \mu(S_j)^2/(2D_j\mu(Q)). Since T_j1_{S_j}\le1, the remaining contribution implies

which proves (4.5); the definition gives (4.6). Conditional on the first j-1 successful transitions, the current state lies in S_{j-1} and the next increment retains its prescribed law. Multiplying the conditional lower bounds proves the claimed probability estimate. For fixed h, finitely many squarings, products, and divisions by e^{o(r)} preserve a lower bound of the form e^{-o(r)}.□

Proof. Apply Lemma 4.2 and choose x_r\in S_{0,r}. Independent sampling of the parameters produces the required independent increment laws, so \nu_r(E_r)\ge e^{-o(r)}. The set E_r is Borel, because \Theta_r, multiplication, and the increment maps are Borel. The density upper bound gives

proving the first conclusion. Lemma 4.1 proves the second. No independence remains necessary after restricting to E_r.□

Proof. On X, the composition bound gives d(\bar F(e),e)\le\Lambda, since F(e)=e. On Y it similarly gives d(F(e),e)\le\Lambda. Equi-Lipschitz bounds now give pointwise precompactness on every compact subset of G. Arzelà–Ascoli and a countable compact exhaustion yield compactness; all the inequalities and composition bounds are closed under compact-open convergence. The same exhaustion metrizes the topology.

The inverse of the first coordinate map is

and that of the second is

They are continuous. Continuity of evaluation and composition follows directly from compact-open convergence and the common Lipschitz bound. The action identities follow either from these coordinates or by substituting the two normalization formulas. For example,

These identities also verify the stated right-action convention.□

Proof. The representation \rho is unitary because \mu is invariant. It is strongly continuous: this follows first on continuous functions from the continuous action on compact X, and then on L^2 by density and unitarity. The jointly continuous evaluation (x,\xi)\mapsto\pi F_\xi(x) is uniformly continuous on compact subsets of G\times X, so b is a continuous L^2 cocycle.

Real-triangulable Lie groups have property \mathcal{WAP}_{\mathrm t}, and hence property H_{\mathrm t}, by [4, Corollary 1.9]. In the form needed here, every continuous unitary representation without invariant vectors has zero first reduced cohomology. Reduced cohomology uses the closure of the coboundaries for uniform norm convergence on compact subsets of the group. The Hilbert representations here are weakly almost periodic, so the cited theorem applies.

Ergodicity identifies the invariant vectors in L^2(X,\mu;W) with the constant functions. Orthogonal projection of (5.1) onto this subspace gives the continuous homomorphism

Apply the vanishing theorem to the cocycle b-l in the orthogonal complement. It gives (5.2). One may complexify the real Hilbert space and take real parts if using the complex convention for unitary representations.

For completeness, fix a compact symmetric generating neighborhood Q such that every x is a product of at most a|x|_G+a elements of Q. A fixed-radius metric ball has this property by subdividing a geodesic. Set

Given \epsilon>0, fix j with \sup_{x\in Q}\|c_j(x)\|_2\le\epsilon. The cocycle identity and unitarity imply

Keep this j fixed, let r\to\infty for |x|_G\le Rr, and then let \epsilon\downarrow0. This proves (5.3). It is a supremum of L^2 norms, which will be sufficient for finite meshes of paths.

A continuous homomorphism to the additive group W kills the derived Lie algebra. Since \mathfrak n\subset[\mathfrak g,\mathfrak g] and N is connected, it kills N; on D it factors through the abelianization V.□

Proof. Fix v_0\in E_V and a pivot d_0\in D of intrinsic length O(r) whose abelianization is rv_0. Put E=\pi E_V. Apply Lemma 3.5 with the set E_V\subset V: although l factors through V on D, we do not yet know that it factors through W. For each n\in B_r(E), that lemma supplies a measurably chosen path from d_0 to nd_0 of length O(r). Its abelianized D-legs stay in rE_V, and its switches have length o(r). Since l kills N and is Lipschitz, the path’s l-heights lie in r(lE_V+o(1)). All its points have distance O(r) from e, with common bounds for the sampled endpoints.

Sample n with normalized Haar measure on B_r(E), independently of \xi\sim\sigma. Fix \tau>0 and sample each path at an arclength mesh of spacing at most \tau r, including its endpoints. There are at most M_\tau sampled points, independent of r,n. At any of them, say g_r(n), the assumed estimate gives

uniformly in n. Fubini, a finite union bound, and Chebyshev’s inequality show that, for every fixed \eta>0, the fraction of pairs (\xi,n) on which a sampled error exceeds \eta r tends to zero. Hence for all large r there is one \xi for which at least half the endpoint volume has no such error.

For these endpoints the entire image path lies at heights within (c\tau+\eta+o(1))r of r lE_V. Indeed the image of each mesh interval has diameter at most the common Lipschitz constant times \tau r. The initial image point is the common F_\xi(d_0); subtracting its height only translates the allowed height set.

At their common source D-position, the good endpoints have a separated subset of cardinality at least c_0 e^{rP(E)}, by Lemma 3.8. Choose the separation large enough that the quasi-isometry lower bound keeps their images separated by a fixed positive amount. Their image paths have length O(r). Lemma 3.6 and translation invariance of P give

after taking r\to\infty. First let \eta\downarrow0, then \tau\downarrow0. Continuity of P for Hausdorff distance proves (5.4). This argument only used a fixed finite mesh before the limit; it did not require an L^2 bound on the pointwise supremum over a path.

Apply (5.4) to [0,v]\subset V. Since p is a norm,

Writing d=\dim W, the inclusion forces \rank l\ge d, while the target of l has dimension d. Thus l and \pi have the same kernel on V, and l=A\pi there for an invertible A. Both homomorphisms kill N and G=ND, so this equality holds on G. All the reasoning applies to the reversed pairs on Y as well.□

Proof. The invariant measure on the inverse section. Let m be Haar measure on G. In the coordinates \Omega=G\times X of Lemma 5.2, put M=m\times\mu. This is a nonzero Radon measure and is target-invariant. Domain translation has the formula

For each fixed \xi, the change in z is right translation, which preserves m by unimodularity. Invariance of \mu therefore makes M domain-invariant as well.

In the other coordinates \Omega=Y\times G, domain translation is (\eta,x)\mapsto(\eta,xg). For a Borel set E\subset Y, the measure B\mapsto M(E\times B) is right-invariant and locally finite: for compact B it is bounded by M(Y\times B)<\infty. Haar uniqueness gives M(E\times B)=\nu(E)m(B), and countable additivity in E supplies a Borel measure \nu with

Here a product with compact factors is compact in \Omega, by the coordinate homeomorphism, so the local finiteness used above follows from the Radon property of M.

Choose a compact ball B_0\subset G with 0<m(B_0)<\infty. Then

If \nu(Y)=0, a countable compact exhaustion of G and (5.7) would imply M=0. Thus 0<\nu(Y)<\infty. In particular, \nu is a measure obtained from product coordinates; we are not restricting M to the possibly M-null section Y.

For y\in G, set q_y(\eta)=\bar F_\eta(y). The identity

shows that \nu is U-invariant. More explicitly, integrate a test function of \eta times a compactly supported function of x against M. In the transformed integral, the fiber change x\mapsto q_y(\eta)^{-1}x preserves left Haar measure. Target invariance of M then gives invariance of the \eta-integral.

Transfer functions averaged over the fibers. Choose measurable representatives of the \phi_j and lift them by

They belong to L^2 on every compact subset of \Omega and satisfy \Phi_j(L_z\omega)=\pi z+\Phi_j(\omega). Their domain increments have defect

If \omega=L_z\xi, this is exactly c_j(t)(\xi)=b(t)(\xi)+\phi_j(T_t\xi)-\phi_j(\xi)-A\pi t. Consequently, for compact K_\Omega\subset\Omega and K_G\subset G,

Indeed K_\Omega lies in K\times X for a compact K\subset G in the first coordinates; the integral is at most m(K)\|c_j(t)\|_2^2.

We now average over whole balls so that no trace of an L^2 function on a null section is required. Fix a compact ball B\subset G of positive Haar measure, and define

Product coordinates and Fubini define this average for \nu-almost every \eta. Jensen’s inequality and local L^2 integrability of \Phi_j show that \Psi_j\in L^2(Y,\nu;W); define it arbitrarily on the remaining null set.

Equation (5.8), target equivariance of \Phi_j, and additivity of \pi give

where R_j(\eta,y) is the difference of the averages of D_j(\eta,\cdot) on q_y(\eta)B and on B. For each fixed y, this identity holds almost everywhere: U_y preserves \nu, and the changes in the fiber variable preserve Haar measure. These facts also show that changing the measurable representatives of the transfers changes none of their L^2 classes or averages.

Comparing two fiber averages. We claim that R_j tends to zero in L^2(\nu) uniformly for y in any fixed compact Q. The increment estimate (5.9) will control, in L^2(\nu), the difference between averages over two bounded regions of each fiber. Since \bar F_\eta(e)=e on Y and the maps have a common Lipschitz constant, all q_y(\eta) with y\in Q, \eta\in Y lie in one compact set Q_1\subset G. Put C_1=Q_1B. Jensen’s inequality applied to the two averages gives, for every y\in Q,

The enlargement from q_y(\eta)B to C_1 is legitimate because its averaging density is bounded by 1/m(B), even though its center depends on \eta. Thus the right side is independent of y.

Inside this integral one has

Make the substitution t=x_2^{-1}x_1, which preserves left Haar measure in x_1. Enlarge the resulting t-domain to the compact B^{-1}C_1. The pairs \eta\cdot x_2 range over the compact image of Y\times B, and their measure is M by (5.7). Hence the right side of (5.11) is at most

which tends to zero by (5.9). We have proved

The supremum here is over the individual L^2 norms. No common pointwise exceptional set for all y is being asserted or used.

Set \theta_j=-A^{-1}\Psi_j and rearrange (5.10). This gives

proving (5.5) with precisely the inverse of the given A. The inverse height evaluations form a continuous cocycle for the unitary action on L^2(Y,\nu;W), by the formulas in Lemma 5.2. Subdividing a geodesic into bounded steps, exactly as in the proof of Lemma 5.3, proves (5.6). Neither ergodicity of \nu nor a second application of reduced cohomology is needed.□

Proof. Lemma 5.3 gives l and its compact coboundary approximations. Lemma 5.4 gives l=A\pi with A invertible. The inverse coupling in Lemma 5.5 supplies a finite nonzero invariant measure \nu and the inverse averaged estimate with the same A^{-1}. Normalize \nu to a probability and apply Lemma 5.4 to the inverse maps.

Every compact convex E\subset W has a compact convex lift to V, by a fixed linear section of \pi:V\to W. The forward comparison therefore gives P(E)\le P(AE), and the inverse comparison applied to AE gives P(AE)\le P(E). Thus P(AE)=P(E) for every nonempty compact convex E, which is precisely A\in\mathcal A.□

Proof. Write \mu=\int\mu_s\,d\kappa(s) for its ergodic decomposition [17, Theorem 5.2]. On almost every component Proposition 5.6 gives A_s\in\mathcal A. Since |\exp_D(Lv)|_G\le L\|v\| up to the fixed norm convention, its estimate implies convergence of the displayed integral with \mu_s in place of \mu.

The normalized height evaluations are uniformly bounded for L\ge1 by the Lipschitz bounds for F and \pi and by F(e)=e. The comparison set is finite and fixed. Dominated convergence through the ergodic decomposition proves (5.12). No measurable choice of A_s is required: the integrand already takes distance to the full finite set. Continuity for fixed L follows from evaluation in the compact-open topology.□

Proof. Expand \log k_0 in the finite list of bracket words using Lemma 3.1. Its coefficient in a word Y is bounded by \exp(r(1+\delta)h_Y(\mathcal C)+o(r)). The finite expansion of \exp(\ad(\log k_0))\log u_r consists of brackets of an initial word Z with additional words Y_1,\ldots,Y_j. The sum of their lengths is at most m whenever the bracket is nonzero. Before expansion in a maximizing basis, the logarithm of the coefficient divided by r is at most

The width of the resulting word is h_Z(\mathcal C)+\sum_\ell h_{Y_\ell}(\mathcal C). Relative to this width the exponent is therefore at most

This is bounded above by a negative constant for all large r. The basis-exchange comparison in Lemma 3.1 says that expanding a word in a maximizing basis uses only basis entries whose widths are at least its width. That expansion cannot increase the bound in (6.6). There are only finitely many terms, which proves the assertion.□

Proof. We prove that the laws of the sampled points are asymptotically unchanged in total variation by right multiplication by each fixed element of G. It suffices to consider elements of D and N, since these subgroups generate G.

For d\in D, right multiplication changes z to

This map preserves Haar measure in the scaled coordinates, because D is unimodular and the coordinate scaling has constant Jacobian. By Lemma 2.2 it converges to the identity uniformly on compact sets. A continuous compactly supported density changes by o(1) in L^1, proving the assertion for d.

For fixed u\in N, right multiplication changes a to

and leaves z unchanged. On the compact support of \rho, the weight-action estimate in Lemma 2.3 bounds a coefficient of \log\Ad_{\Delta_r z}(u) in a word Z by

The error is uniform there; it includes the polynomial factors from the unipotent part of the D action. Words of zero total weight cause no exception to the inequality. Lemma 6.2 shows that q_r(z) tends exponentially to the identity in the \mathcal C-normalized coordinates, uniformly in z.

The normalized product laws have uniformly bounded polynomial coefficients. Thus, on the fixed coordinate cube defining B_r(\mathcal C), multiplication by q_r(z) and by q_r(z)^{-1} both converge uniformly to the identity. Their images lie in Euclidean boundary enlargements of the cube of thickness tending to zero. Haar invariance now gives

This is precisely the required total-variation estimate for the conditional uniform laws of a.

The estimates were proved before projecting the sampling coordinates to G. Projection decreases total variation, so the possible nonuniqueness of ND coordinates is harmless here. Finally apply (6.7). The action on the compact space X is continuous by Lemma 5.2; every weak limit of the normalized occupation laws is therefore invariant.□

Proof. First allow the starting point z to vary in a small neighborhood O so that the whole family of segments lies in a compact subset of \mathcal U_\delta. Choose a continuous probability density \eta supported in O, positive near the specified starting point. Choose another continuous probability density \rho, compactly supported in \mathcal U_\delta, which is bounded below by a positive number on a compact neighborhood of this swept family. The same neighborhood contains all the actual scaled paths \Psi_{r,t}(z) from (6.5), for all large r, z\in\supp\eta, and 0\le t\le T.

Let \mu_r be the occupation law obtained with \rho. Pass to a subsequence on which \mu_r converges to \mu; the prescribed profile limit remains \nu. By Lemma 6.3, \mu is invariant. The cutoff and this subsequence are fixed before choosing a microstep length.

Fix an integer L\ge1. Put q=\lfloor rT/L\rfloor and divide the actual path into q increments by \exp_D(Lv) and a final remainder. For 0\le j<q, its jth starting point is

Right invariance of Haar measure in D shows that the distribution of the scaled D coordinate of p_j, when z has density \eta, has density at most \|\eta\|_\infty. Its support is in the compact set where \rho has a positive lower bound. Hence the law of \omega[p_j] is bounded above by C_0\mu_r, with a constant C_0 independent of r,j,L. The original law of a is unchanged and independent of z throughout this comparison.

Write K_v=\operatorname{conv}\{A\pi v:A\in\mathcal A\} and define the bounded continuous function on X

The identity of height cocycles at p_j and convexity give

The final error uses the Lipschitz bound on the remainder interval and the boundedness of K_v. For this fixed L, let r\to\infty. Profile compactness and the uniform convergence of \Psi_{r,T} give

Now let the integer L tend to infinity. Corollary 5.7 for this one fixed invariant probability \mu makes the right side tend to zero. No convergence rate uniform over invariant probabilities is required.

The nonnegative integrand on the left is continuous in the profile and the starting point. Its integral being zero, it vanishes on the product of the supports. In particular it vanishes for the specified f and z. The argument applies to each segment as stated in the lemma.□

Proof. Fixing the endpoint scheme. Set E=\pi J. We may pass to any further subsequence of the one giving \nu. First pass to a subsequence on which the nilpotent laws in E-normalized coordinates converge. By Lemma 3.3, choose a fixed finite ordered list of generator increments whose normalized endpoint map has a submersive limit at an interior parameter point. Its length h is fixed independently of r. An increment in letter i has the form \exp(t e^{rs_i(E)}X_i), with t in a fixed bounded interval. Replace t by t_1-t_2 with independent uniform interval parameters, enlarging their fixed intervals if necessary. The difference map is surjective, so the limiting endpoint map is still submersive somewhere in the parameter cube. Fix this scheme now.

Retaining sheets with the prescribed profile. Choose a compact neighborhood Z_J\Subset\mathcal U_\delta of J that contains all the actual scaled paths \Psi_{r,t}(z), 0\le t\le T, for large r. Put Q_r=B_r(\mathcal C)k_0. Because f\in\supp\nu, for each fixed positive tolerance the set of n\in Q_r whose profiles are within that tolerance of f on this compact has a positive lower limiting relative measure. Choosing the tolerances slowly enough gives Borel sets S_r\subset Q_r and numbers \zeta_r\to0 such that

To be explicit, choose tolerances 1/j successively; postpone the jth choice until its probability lower bound has logarithm of absolute value at most r/j. This also ensures that every other fixed compactness error tends to zero along the diagonal. The scheme and its compact were chosen before this diagonal.

We verify the support-volume hypothesis for recurrence. For an individual generator interval let \sigma_{j,r} denote the law of \exp(t e^{rs_i(E)}X_i). Since E\subset(1-c\delta)\operatorname{int}(\mathcal C),

Lemma 6.2 shows that k_0\supp(\sigma_{j,r})k_0^{-1} tends to the identity in normalized \mathcal C coordinates. The bounded polynomial product law therefore gives

All estimates hold with common envelopes for the fixed finite list.

For the jth switch, sample s_{j,1},s_{j,2} independently with law \sigma_{j,r}, independently also for different j, and put u_j=s_{j,1}s_{j,2}^{-1}. Its law is \sigma_{j,r}*\check\sigma_{j,r}. Apply Lemma 4.2 in the unimodular group N with the compact set Q_r, the Borel set S_r, and these increment laws. The mass bound (6.10), the denominator bound (6.11), and the fixed depth h give a starting point n_0\in S_r such that

Thus every intermediate sheet in a successful sequence has a good profile.

From parameter measure to endpoint volume. The successful interval parameters have Lebesgue measure at least e^{-o(r)} in the fixed product cube, since the uniform product density is a fixed constant. In E-normalized logarithmic coordinates, the endpoint map for the increments is a polynomial of fixed degree with convergent bounded coefficients and a submersive limit, by our choice of scheme. Lemma 4.1, or equivalently Corollary 4.3, now gives accessible endpoint volume in N at least

If necessary, restrict the successful parameters to a compact subset of half their measure; this preserves the estimates and makes the image measurable. Left multiplication by n_0 preserves the volume. The rank conclusion is a property of the full fixed polynomial map, established before the successful restriction.

Comparing source and image packing. Take d_p=\Delta_r z as pivot. For each prescribed generator increment, travel along \Delta_r z\exp_D(rt v) to a point where its weight attains s_i(E), perform the increment, and return to the pivot on the new sheet. Lemma 3.5 gives an o(r)-length switch at that point; the two D legs have length O(r). The difference of two interval parameters is a single generator increment. We therefore do not need a good profile at the algebraic midpoint of s_1s_2^{-1}.

All sheet states at which a D leg is traversed belong to S_r. The images of these legs consequently have heights in

The short switches contribute only o(r) height errors by Lipschitzness. The finite concatenation has length O(r), begins at xn_0d_p, and ends at the fixed D position d_p on one of the accessible sheets.

By Lemma 3.8, the endpoint volume (6.12) gives at least \exp(rP(E)-o(r)) source points separated by any prescribed fixed large constant. Choose this constant so their quasi-isometry images remain separated. Their image paths have a common start, length O(r), and relative heights in r(f(J)-f(z)+o(1)), since the starting profile value is f_{r,x,n_0}(z)=f(z)+o(1). The compact set f(J)-f(z) contains zero; enlarging it by a ball of radius tending to zero absorbs the error. Lemma 3.6, translation invariance of P, and its continuity for Hausdorff distance give the upper bound \exp(rP(f(J))+o(r)) for the same separated points. Taking logarithms and dividing by r proves (6.9). When \pi v=0 the claimed lower bound is zero; it is also immediate from nonnegativity of P.□

Proof. Horizontal derivatives take values in the finite group. Choose a basis of the first layer V and its left-invariant vector fields on D_\infty. Flow-box Fubini and linewise Lipschitzness give locally bounded distributional derivatives of f in these directions. Their linear combinations define a measurable linear map M(z):V\to W. For each fixed v, it agrees almost everywhere with the line derivative in direction v: both are the distributional derivative along that field. These assertions may be made simultaneously for a countable dense set of v by intersecting their full Haar-measure sets.

At a point of line differentiability, the image of a short segment, translated to start at zero and divided by its length, converges in Hausdorff distance to the linear segment of velocity M(z)v. Indeed the definition of differentiability bounds the error uniformly on the entire sufficiently short one-sided interval. Differentiate (6.13) and use continuity and positive homogeneity of P to obtain

at the common full-measure points. Continuity in v extends these constraints from the dense set to all of V.

The first constraint makes M vanish on \ker\pi, so M=B\pi for a linear map B:W\to W. It also gives p(Bw)\le p(w) by convexity, while the second constraint gives the reverse inequality. Thus B is a linear isometry of p and is invertible. The full linear isometry group of the polyhedral norm p is finite: its elements permute the finitely many vertices of the unit ball, and those vertices span W. Average a Euclidean inner product over this full finite group. In its norm, Bw and every Aw, A\in\mathcal A, have the same length. Strict convexity and the convex-hull inclusion force

If B differed from every A\in\mathcal A, the finitely many proper linear subspaces \ker(B-A) could not cover W. Hence B\in\mathcal A. We have proved

Choose any label on the remaining null set, so B is a measurable finite-valued function throughout the domain.

Equality of support and variation fixes weight signs. Let v have generic height w=\pi v, meaning \alpha_i(w)\ne0 for all i. Along almost every line of this direction, the absolutely continuous restriction h(t)=f(z\exp_\infty(tv)) has derivative B(z\exp_\infty(tv))w almost everywhere. Thus p(h'(t))=p(w). We work on a compact interval [0,T] inside such a line and translate h to have h(0)=0, which does not change P.

For a word basis \mathcal B, let m_i(\mathcal B) count all occurrences of X_i in its words. Every generator is mandatory, so m_i(\mathcal B)\ge1. Choose a basis maximizing P(h([0,T])), and put q_i(t)=(\alpha_i\circ h)'(t). Then

The second inequality is the definition of p as the maximum over word bases. The segment volume inequality supplies the reverse inequality between the outer quantities, so every inequality in (6.16) is equality. Since every m_i(\mathcal B) is positive, each scalar maximum has no deficit. If t_i is a maximizing time for \alpha_i(h(t)), this says

Consequently a negative derivative cannot occur, on a set of positive measure, before a positive derivative. On this interval each weight’s positive-sign indicator is nonincreasing, up to null sets.

By Lemma 3.2, pullback by every label permutes the distinct oriented generator-weight rays. Genericity therefore makes every q_i nonzero almost everywhere, and the number of positive signs among those distinct rays is independent of the label B. A finite collection of nonincreasing indicators with constant sum has each indicator constant almost everywhere. Thus all weight signs are constant along the line, on the interval under consideration. Applying this on a countable exhaustion of its open interval in \mathcal U_\delta shows the following: if w lies in a chamber of the generator-weight hyperplane arrangement, then almost all labels along the line send that chamber to the same chamber. Here each line meets \mathcal U_\delta in an interval because its projected height moves linearly through a convex set.

Extremal directions have constant horizontal derivative. The chambers just used are the connected components of W\setminus\bigcup_i\ker\alpha_i. Their closures are pointed polyhedral cones, since the weights span W^*. Let w\ne0 span an extremal ray R of one of these closed cones, and choose a horizontal lift v\in V.

For each chamber C_0 whose closure contains R, choose generic vectors v_j\to v with \pi v_j\in C_0. We explain carefully why the almost-everywhere chamber agreement passes to this limiting direction. On a compact set of pairs (z,s) whose flows stay in the domain, the maps z\mapsto z\exp_\infty(sv_j) preserve Haar measure and converge uniformly to the corresponding map for v. Approximating the bounded measurable function B by continuous functions on a larger compact set shows that

The approximation errors are uniformly controlled by Haar invariance. Apply the same statement at a second time t; Fubini gives convergence in measure locally in (z,s,t). After a subsequence both labels converge almost everywhere. We restrict to compact sets of triples for which the segment between the two limiting positions stays in the open domain; for large j the nearby segments do so too. The preceding chamber argument applies to every v_j. Since the set of labels is finite, the relation that two labels have the same image of C_0 is closed. It follows that almost every such triple satisfies

There are only finitely many incident chambers. Intersect their full-measure sets, and then exhaust the domain of triples by compact sets.

The intersection of the closures of all chambers incident to R is exactly R. Indeed the active hyperplanes of an extremal ray cut down to its line; the remaining chamber inequalities select the half-line containing w. Equivalently, all hyperplanes through w allow both neighboring signs among the incident chambers, while the other hyperplanes have the fixed sign of w. Intersecting the displayed equalities therefore shows that the two labels have the same image of R. They preserve the invariant Euclidean norm chosen above, so their values on w are equal. Thus the derivative in direction v is constant at almost every pair of positions on almost every such line. Absolute continuity makes f affine on those lines.

When v\in\ker\pi, the first inequality in (6.13) already makes f constant on every horizontal segment of direction v.

Regularity makes the label constant on the domain. The extremal rays of a full-dimensional pointed chamber span W. Choose finitely many lifts of such rays spanning a complement to \ker\pi in V, and add a basis of \ker\pi. The resulting vectors v_1,\ldots,v_b span V. Let Y_j be their left-invariant horizontal fields on D_\infty. The line affinities just proved imply

For completeness, use flow coordinates for Y_j and test against a compactly supported smooth function. Along almost every flow line the second distributional derivative of an affine function is zero. Integration in the transverse coordinates gives the displayed identity. The flow preserves Haar measure, so no divergence term occurs.

The associated graded Lie algebra of D is generated by its first layer V. Hence the fields Y_j and their iterated brackets span the full tangent space everywhere. Hörmander’s sum-of-squares theorem [20, Theorem 1.1] applies to \sum_{j=1}^bY_j^2, componentwise in W, and makes f smooth. All hypotheses are local on the open domain: the fields are smooth and real, the bracket condition holds at every point, and the right side is the smooth function zero.

The horizontal derivative is now continuous. Its almost-everywhere membership in the finite set \{A\pi:A\in\mathcal A\} from (6.15) extends to every point, since a full-measure subset is dense. The domain is connected, so that derivative is one constant A\pi. Every horizontal derivative of f-A\pi vanishes; the same is then true for iterated brackets. Bracket generation makes its entire differential zero, and connectedness gives f-A\pi=b for one constant b\in W.□

Proof. Relative compactness and horizontal Lipschitz regularity were proved after Definition 6.1. Lemmas 6.4 and 6.5 supply both assumptions of Lemma 6.6 on every segment. That lemma gives (6.18).□

Proof. For a continuous function on Z, its distance in the supremum norm to the set of restrictions of A\pi+b is a continuous, 1-Lipschitz function of that continuous function. The affine restriction set is closed: take a subsequence with fixed A, and evaluate at one point of Z to see that the offsets converge.

If (6.19) failed, there would be a sequence of maps, basepoints, and deterministic allowed suffixes for which the displayed probability is bounded below by a positive constant. Profile compactness supplies a subsequential law limit. Proposition 6.7 supports that limit on the affine class. The event whose probability is bounded below is closed, so the Portmanteau inequality is a contradiction.

For an independent random suffix, condition on its value. The bad probability is an average of deterministic bad probabilities, and hence is bounded by the same supremum, which tends to zero. For a compact exhaustion of \mathcal U_\delta, the zero sets of the continuous affine-distance functions are closed. For each compact in the exhaustion and each \eta>0, the Portmanteau inequality for the open event that this distance exceeds \eta shows that its limiting probability is zero. Thus every limiting law is supported on functions affine on all these compact sets. Use a constant-label subsequence and evaluate at a fixed point to see that their labels and offsets are compatible across the exhaustion; equivalently, the class of functions affine on the entire domain is closed in C_{\mathrm{loc}}(\mathcal U_\delta,W).

The function \varepsilon(r) was fixed before taking the limit. This assertion is not a supremum over all possible subexponential envelopes at one finite scale. Nor does it assert concentration near one particular pair (A,b): different affine profiles may still occur in the limiting law.□

Proof. Choose one lift for each endpoint of each tuple. Thicken each total lift (U_1\cdots U_j,d_j) by multiplying it on the right by a fixed small compact neighborhood in N\rtimes D. Its projected diameter is uniformly bounded, because q is a homomorphism. The resulting product neighborhoods in (N\rtimes D)^h are disjoint: their projections are contained in disjoint neighborhoods of the separated tuples. Each has the same positive Haar volume. Here product Haar on N\rtimes D is both left and right invariant, by unimodularity and the determinant-one D action on N.

All their D coordinates lie in a ball of radius O(r), of polynomial Haar volume. Fubini therefore shows that their projection to the total N coordinates has volume at least M_r divided by a polynomial in r. The map

preserves product Haar: successively, each changed variable undergoes a left translation. Apply it to this projected set to obtain \mathcal T_r.

To check its bounds, write a thickened total N coordinate as (U_1\cdots U_j)\eta_j. Right multiplication of its full lift by a bounded factor makes \eta_j=\Ad_{d_j}(a_j) for a_j in a fixed compact subset of N. The small height and polynomial unipotent bound give \|\log\eta_j\|\leq e^{o(r)}. The new increment is \eta_{j-1}^{-1}U_j\eta_j, with \eta_0=1. Since 0\in E_j, all word widths of E_j are nonnegative, and the box product estimates put this new increment in B_r(E_j,e^{o(r)}). Every projected total coordinate has at least one accompanying D tuple from the thickened set; this proves the assertion about terminal products as well as (7.1).

For completeness, consider two admissible pairs (u,d) and (\widetilde u,\widetilde d) whose projections lie in the same ball of fixed radius. Lift their bounded group difference as q(a,b) with a\in N and b\in D in fixed compact sets. Equality in G gives

The first factor has log norm at most e^{o(r)}. The second has D-length O(r) and hence polynomial log coordinates, by Lemma 2.2. The logarithm of their product is therefore bounded by e^{o(r)}. Once one admissible pair is fixed, all the other N coordinates in the ball lie in a left translate of an N log ball of radius e^{o(r)}. Its Haar volume is e^{o(r)}. This is also the specific instance of Lemma 3.9 needed here. Notice that (7.2) bounds the entire overlap, without assuming it is central or discrete.□

Proof of Proposition 7.1. Both profiles are covered by Proposition 6.7, since multiplying the allowed suffix by k preserves its enlarged box bound. Suppose the asserted equality of labels fails for arbitrarily small positive \delta. We explain first the limiting geometry and then compare source and target volumes.

A reflection at a facet. Choose violating \delta\to0. Passing to a subsequence fixes the letter X_i and distinct labels A_1,A_2\in\mathcal A. Compactness of the profiles allows their offsets to converge to b_1,b_2. Diagonalize in the scales so that there are Borel sets

on which the two profiles converge to A_1\pi z+b_1 and A_2\pi z+b_2 on every required compact set over \mathcal C. To justify the mass assertion, fix each approximation tolerance first. A neighborhood of a point in a limiting support has positive probability. Take its scale sufficiently large that the negative logarithm of this probability divided by r is arbitrarily small, then decrease the tolerance. A compact exhaustion permits any later fixed finite collection of paths. The domains \mathcal U_\delta exhaust the inverse image of \operatorname{int}\mathcal C; equicontinuity on compact subsets of D_\infty extends this convergence to the boundary along the same diagonal. Also k_0,k\in B_r(\mathcal C,e^{o(r)}) on this diagonal.

At a height in the facet

the points on the two sheets are a distance o(r) apart: conjugate the step by a D position at that height and use Lemma 3.4. Thus the limiting affine maps agree on the relative interior of the entire facet. Their linear parts satisfy

Choose an inner product invariant under the finite group \mathcal A. An orthogonal map fixing a hyperplane pointwise is either the identity or its orthogonal reflection. Since A_1\ne A_2, the map A_1^{-1}A_2 is this reflection.

Let p_0 be the contact point in the relative interior of H_i and let v be its reflection normal, chosen with \alpha_i(v)>0. For a sufficiently small fixed \epsilon>0 put

The common affine value at p_0 is the target pivot. Translation invariance and segment homogeneity of P give

Figure 1 depicts these source and target segments.

Source bundles with a large joint volume. At a pivot d_p\in D of height rp_0 and D-length O(r), alternate two sorts of bundles. A first-sheet bundle updates the pair state by n\mapsto nu. A second-sheet bundle updates it by n\mapsto nkuk^{-1}; on that sheet this is the generator movement nk\mapsto nku. Each bundle consists of a fixed finite sequence of generator increments with widths e^{rs_j(E)} and with independent difference-of-interval parameters before restrictions are imposed.

We choose the source schemes and their number before restricting parameters to good sheets. Pass to a subsequence on which the normalized bracket laws for E, E'_1, E'_2, and E'_1\cup E'_2 converge, together with the bounded linear embeddings from either individual target normalization into the union normalization.

The number of bundles is determined by multiplication in the target. Consider the polynomial map multiplying alternating increments from the two target boxes, with output normalized to E'_1\cup E'_2. For a sufficiently large fixed number h of increments, its limiting map is submersive. Indeed, every union-scaled generator attains its support maximum in one of the two sets. Its limiting vector therefore belongs to the image of the corresponding limiting embedding. These vectors bracket-generate the union-normalized limiting algebra. Finite products of exponentials from the two embedding images have a submersive endpoint map: their conjugates span the generated Lie algebra, and one inserts variable increments with those conjugates as differentials. Inserting identity factors makes the product alternating. Fix such an h. This rank property belongs to the full multiplication map, before any target increments have been selected.

For each of the h source bundles, Lemma 3.3 supplies an E-normalized generator scheme with a submersive limiting product map. Represent its scalar parameters as independent differences of interval parameters. The schemes and their total number of switches are now fixed. These choices depend only on the normalized algebra data, not on S_r or the support tolerances. Passing to this subsequence preserves the profile approximations and the bound \vol S_r/\vol Q_r\geq e^{-o(r)}. All the scaled D-legs lie in one compact subset over \mathcal C, including their facet pivots. The boundary convergence just established therefore controls their profiles.

Apply backward recurrence to the individual switches in these bundles. For a first-sheet switch with interval law \sigma, the increment law is \sigma*\check\sigma. For a second-sheet switch it is \sigma_k*\check\sigma_k, where \sigma_k=(\Ad_k)_*\sigma is the conjugated interval law. Thus all switches have the form required by Lemma 4.2, with a depth independent of r.

Every recurrence denominator in Lemma 4.2 is at most e^{o(r)}\vol Q_r. Indeed the supports in question are products of a fixed number of k_0^{\pm1}, k^{\pm1}, and generator intervals fitting the \mathcal C widths, all with subexponential factors. The box calculus bounds the volume of the corresponding support enlargements by e^{o(r)}\vol B_r(\mathcal C). Hence backward recurrence supplies a nonempty Borel subset of S_r such that each initial state in that subset has conditional successful parameter probability at least e^{-o(r)}, with every intermediate pair state in S_r. Fix one such state n_0.

Apply Corollary 4.3 to the map from all bundle parameters to the tuple of their products before the conjugations by k and before accumulation. The normalized map is the product of h submersive limiting polynomial maps. The normalized pairs of interval parameters have a fixed bounded joint density with respect to Lebesgue measure. Thus the successful probability just proved gives successful Lebesgue measure at least e^{-o(r)}, even though the restricted coordinates are correlated. The image consequently contains a compact set of N^h Haar volume at least e^{rhP(E)-o(r)}. Conjugation by k preserves Haar on N, and the change from successive increments to successive states is triangular and Haar preserving. Consequently the admissible tuples (n_1,\ldots,n_h) of bundle-end states have the same lower volume. This application takes place before conjugation, so it requires no coefficient bound for a conjugation written in E-normalized coordinates.

By Lemma 3.8, the corresponding tuples

have a separated-count lower bound e^{rhP(E)-o(r)}. Take the source separation sufficiently large that the quasi-isometry preserves a fixed target separation. On the source side each terminal point xn_hd_p is reachable from xn_0d_p by a path of length O(r) whose relative heights lie in r(E_0+o(1)). Use the D legs over E and return to the pivot after each generator switch. Second-sheet bundles use o(r) bridges at the pivot. Lemma 3.6 therefore bounds the covering number of all source terminal points by

at every fixed radius.

Target products have twice the terminal exponent. Put y_j=F(xn_jd_p). The successful profiles imply that the transition from y_{j-1} to y_j has a path of length O(r) with relative height set r(E'_{\tau(j)}+o(1)), where \tau(j) alternates between 1 and 2. All pivot height differences are o(r), since the two limiting affine profiles agree at p_0 and every intermediate state is good. Choose lifts of these paths. On accumulating them, the preceding D factors have height o(r) and length O(r); their conjugations preserve the indicated box widths up to subexponential factors. The endpoint lifts thus have the form

Apply Lemma 7.2 to the separated target tuples. It gives a measurable increment set of Haar volume at least e^{rhP(E)-o(r)}. Dividing each factor by its box determinant shows that this set has Lebesgue measure at least e^{-o(r)} in the joint individual-box log coordinates. These coordinates lie in a cube of radius e^{o(r)}.

Apply the alternating multiplication map used to choose h, with output normalized to E'_1\cup E'_2. Its degree is fixed, its coefficients are bounded by the box calculus, and its limit is submersive by that earlier choice. These properties concern the full map and therefore remain available for the measurable increment set just obtained.

Lemma 4.1, applied to the measurable increment set, gives a compact set of terminal N coordinates of volume at least

Every such terminal coordinate has an accompanying D coordinate of length O(r) and height o(r) projecting near an image terminal. The last assertion of Lemma 7.2 bounds its volume over any fixed-radius projected ball by e^{o(r)}. But (7.4) and the quasi-isometry cover all image terminals by only e^{rP(E)+o(r)} fixed-radius balls. This is a contradiction because P(E)>0. The labels therefore agree for every sufficiently small fixed \delta, as asserted.□

Proof. Bracketing a mixed word with any generator produces another mixed word. The derivation identity and the Jacobi identity therefore show that their span is the ideal generated by the outside letters. Homogeneous bracket words span \mathfrak n. If a weight \gamma lies outside S, a word of weight \gamma cannot be pure: a sum of inside weights belongs to S. Consequently the entire generalized weight space \mathfrak n_\gamma lies in \mathfrak i_S whenever \gamma\notin S. For v\in\mathfrak d and an outside letter X_i, the vector [v,X_i] stays in \mathfrak n_{\alpha_i} and hence in \mathfrak i_S. Applying the derivation identity to words proves [\mathfrak d,\mathfrak i_S]\subset\mathfrak i_S. This argument includes the nilpotent parts of the generalized weight actions.

Both possible cones S are pointed. A nonempty sum of their nonzero inside weights cannot be zero. The quotient algebra is spanned by pure words, so it has no zero generalized weight space. The zero weight space of \mathfrak n is therefore contained in \mathfrak i_S; in particular \mathfrak d\cap\mathfrak n\subset\mathfrak i_S. The exponential descriptions of the subgroups give N\cap D\subset I_S. The ideal exponentiates to a closed subgroup because the exponential map is a diffeomorphism and \mathfrak i_S is a closed linear subspace. The Baker–Campbell–Hausdorff group on \mathfrak n/\mathfrak i_S identifies with N/I_S: the algebra quotient induces a surjective group homomorphism with kernel I_S. This realizes L_S as a connected, simply connected nilpotent Lie group. A linear section of the algebra quotient, composed with its logarithm and the exponential map of N, gives a continuous section L_S\to N. Images of compact sets under this section are compact, proving the lifting assertion.

The action of N\rtimes D on L_S is

The kernel of the multiplication map N\rtimes D\to G consists of (h^{-1},h) with h\in N\cap D. Its action sends xI_S to xh^{-1}I_S=xI_S. Thus the action factors through G, without requiring that the multiplication map be injective. If g=nd, its value at the identity is nI_S, proving the stabilizer assertion and the stated equivariance of p_S.

Translations by N preserve quotient Haar measure. The determinant of the D-action is the exponential of its trace character; its unipotent parts have determinant one. The weight description thus gives (8.1). To check its positivity assertion, project to the abelianization of \mathfrak n. The image of \mathfrak i_S there is exactly the span of the outside generator images. The inside generator images remain linearly independent in the quotient abelianization. They can be included in a basis of \mathfrak n/\mathfrak i_S selected from pure words. Summing that basis’s weights proves the asserted integral combination. Every inside weight in a two-generator cone is a nonnegative real combination of its two defining weights, and each defining generator itself survives. Both extremal coefficients are therefore positive.□

Proof. If Z is mixed, adjoining occurrences of X_i does not change its width on T_*, because s_i(T_*)=0. In

every nonzero term is mixed and has the same c_Z. Its added exponential cost is at most pM\delta r+o(r). Expanding a word in a maximal-width ideal basis uses only basis vectors whose widths are at least its own, by basis exchange. Consequently the positive differences

absorb all those costs if \delta is sufficiently small. There are only finitely many words and p<m. Their positive gaps also absorb the common o(r) envelope. We obtain

Put Q_r=k_qB^I_r(T_2,1)k_q^{-1}. Choose a maximal disjoint family i_\kappa Q_r with anchors in B^I_r(T_1). Such a family is finite: each set lies in B^I_r(T_1,O(1)), by the first inclusion and normalized multiplication. Conjugation by k_q preserves Haar on I_S, since its derivative is the exponential of the nilpotent map \ad(t_qX_i)|_{\mathfrak i_S} and has determinant one. Dividing volumes in (8.3) bounds the number of anchors by O(e^{r(P_1-P_2)}). Maximality implies that every translate iQ_r, for i\in B^I_r(T_1), meets some i_\kappa Q_r. Hence

Assign a point to the first set containing it. This gives the measurable partition, and the second inclusion above proves (8.7).

For the last assertion write u=i\exp_N\sigma(s), where i\in B^I_r(T_2,e^{o(r)}) and \|s\|\le e^{o(r)}, using (8.5). Factor

The polynomial coordinate maps show that \|\log i_{e_r}\|+\|s_{e_r}\|\le e^{M'\delta r+o(r)} for a fixed multiple M' of M. Every ideal basis width on T_2 is strictly positive. Decrease \delta so that M'\delta is less than all these widths. Then i_{e_r}\in B^I_r(T_2,O(1)) for all sufficiently large r, and the ideal coordinate ii_{e_r} has the required bound. Finitely many simultaneous requirements have a positive common threshold for \delta.□

Proof. Take \delta below the threshold in Proposition 7.1. The two profiles in a limiting support then have the form

with one A\in\mathcal A. Fix such a pair and a subsequence whose joint laws converge to a law containing it in its support. It suffices to prove b_1=b_2. There are finitely many letters and labels, so these may be fixed after passing to a subsequence. Write \alpha=\alpha_i.

Choice of templates. We will use the ray cone of \alpha, and then each cone generated by \alpha and a generator weight independent of it. Fix one of these cones S, and put S'=(A^*)^{-1}S. Let p_0 be the contact point of the unit ball with the \alpha facet of \mathcal C. This point belongs to the relative interior of that facet. Choose a fixed c'>c and put

Choose fixed radii 0<\epsilon_2<\epsilon_1<\epsilon_+ as above, small enough that

for every sufficiently small fixed \delta. Here \Subset means compact containment. To verify the choice, the other oriented-ray facets have positive slack at p_0; a sufficiently small fixed \epsilon_+ preserves their inequalities. On the \alpha facet, \alpha(T_j)\le0, while c'>c gives a strict margin at w_q. The radii can therefore be chosen before \delta tends to zero. On the target templates AT_j the inside weights of S' have width zero and all mixed words have positive width. Also

The comparison will balance three counts. We will construct at least e^{rP_1-o(r)} separated source endpoints and assign them to at most e^{r(P_1-P_2)+o(r)} tiles. For each tile we will prove an upper bound

for the separated image endpoints. The term P_2 comes from their ideal coordinates; the remaining term is the Haar multiplier on the quotient. Summing over tiles cancels P_2 and will force \ell_{S'}(b_2-b_1)\le0. The construction below arranges these two coordinate restrictions on the same endpoints.

Normalize the source at w_q using the exact automorphism S_{-rw_q}, and write k_q=S_{-rw_q}k. Since \alpha(p_0)=s_i(\mathcal C),

Use Lemma 8.2 to partition B^I_r(T_1), with anchors i_\kappa. Choose a fixed unit cube J in \mathfrak n/\mathfrak i_S, and write \sigma for the section used in (8.4).

Three paths chosen independently of the starting sheet. Sample i_1 uniformly in B^I_r(T_1) and s uniformly in J, independently of the original random a. Let i_\kappa be the anchor assigned to i_1, so the tile index \kappa depends only on this probe. For the moment n=ak_0 is still random. Define

We will count the endpoints n_1. The auxiliary endpoint n_2 keeps the ideal probe i_1 and replaces the quotient factor by the neighbor step k, while n_{2,\kappa} gives one second-sheet anchor for its tile. Their design gives the exact cancellation

This difference is independent of i_1. A path from n to n_1 will control the target quotient coordinate relative to the first sheet. A path from n_{2,\kappa} to n_2, followed by the bridge in (8.10), will control the target ideal coordinate of the same endpoint relative to its second-sheet anchor.

The following three sheet changes can be generated with uniformly bounded numbers of generator switches:

For the first, i_1\exp_N\sigma(s)\in B_r(T_1,O(1)) by (8.5). For the second, the relative normalized increment is k_q^{-1}i_\kappa k_q, which satisfies (8.7). For the third it is k_q^{-1}i_\kappa^{-1}i_1k_q\in B^I_r(T_2,O(1)) by (8.6). Lemma 3.3, followed by S_{rw_q}, gives the stated generation with generator amplitudes at most e^{rs_j(w_q+T_l)+o(r)} for the indicated template l.

Choose these product representations measurably as functions of (i_1,s), using the measurable bounded product selection in Lemma 3.3. Its finite family of bounded schemes can be padded by identity factors, and its Borel local inverse choices give Borel parameters for every endpoint. The anchors are already measurable. All choices, including the product parameters and the paths below, are independent of a. The number of switches is bounded independently of the probe parameters. Every intermediate suffix is a fixed-length product of the original allowed suffix and increments in the template boxes, and, when needed, k. The box calculus and (8.8) keep it in B_r((1+\delta)\mathcal C,e^{o(r)}), with one envelope for all probes.

Choose a pivot d_q\in D with \pi d_q=rw_q and |d_q|_D=O(r), for example by exponentiating r times a fixed linear lift of w_q to \mathfrak d. Each generator switch can be performed by a D-leg from the pivot to a support height of its template, the short switch there, and a reversed D-leg on the new sheet. The legs have length O(r) and the switches have length o(r), by Lemmas 3.4 and 3.5. Their scaled D-coordinates lie in a fixed compact subset of \mathcal U_\delta, by Lemma 2.2 and the strict containment (8.8). We fix a compact neighborhood of all these coordinates for the remaining profile comparisons.

Why the intermediate profiles retain their starting offsets. This step uses only the affinity of individual limiting profiles. Consider one of the selected generator switches, with template E\Subset(1-c\delta)\operatorname{int}\mathcal C and letter X_j. There is a nonempty open set of heights in that domain on which \alpha_j(w)>s_j(E), because the support maximum is attained inside the open domain. On a smaller such open set the difference is bounded below by a positive number. At its D-positions, conjugating the switch back gives exponentially decaying generator coefficients, up to polynomial factors. The distance of its two source sheet points, divided by r, therefore tends to zero. Their image profiles agree there in every joint limit. Proposition 6.7 makes each limiting profile affine on \mathcal U_\delta, so equality on this open set gives equality throughout that domain.

It follows by sequence compactness that the difference across each such switch tends to zero in probability on the fixed compact set of legs. This statement is uniform for the permitted parameter choices with their common envelope: a violating sequence of choices would give a joint limit contradicting the preceding open-set argument. Conditioning on any independent probe parameters and integrating proves the same assertion for the measurable choices just made. Since there are boundedly many switches, with probability tending to one their intermediate profiles all differ by o(1) from the starting profile of their respective sheet. The first path uses the profile of n; the other two use that of nk.

We now choose a single starting n retaining a positive volume of probes. For a fixed tolerance \eta>0, let E_r(\eta) be the event that the original pair of profiles is within \eta of (h_1,h_2) on the required compact. The support hypothesis gives \liminf_r\mathbb P(E_r(\eta))>0. Let b_r(\eta) be the joint probability, over the original chart sample and the independent probes, that one of the intermediate profile comparisons has error exceeding \eta. The preceding paragraph gives b_r(\eta)\to0. For large r it is less than one quarter of \mathbb P(E_r(\eta)). Fubini’s theorem then supplies an original sample in E_r(\eta) for which fewer than half of the probes fail these comparisons. First do this for each fixed tolerance and then let the tolerance decrease sufficiently slowly with r. We obtain a deterministic starting n=n_r and a measurable set of good probes of probability at least 1/2, on which all the required intermediate profiles approach their designated affine maps. This order of choices does not condition on a null event, and it does not require every probe to succeed simultaneously.

The generation selectors prescribe the paths; the probability law here remains the original product of ideal Haar measure and Lebesgue measure on J. The preceding success events are measurable because the selected parameters are Borel and the profiles depend continuously on their sheet coordinates. Inner regularity of this finite-dimensional product measure allows us to retain a compact subset of good probes of probability at least 1/4.

The map (i_1,s)\mapsto i_1\exp_N\sigma(s) is injective and carries product Haar measure to Haar measure up to a fixed scalar. The good probe set thus has actual N-volume bounded below by a fixed positive multiple of e^{rP_1}. Since S_{rw_q} has determinant one and left translation by n preserves Haar, the good values of n_1 in (8.9) have this same lower bound. The possibly small probability of the starting event has caused no loss of probe volume. We use the compact retained subset from now on. Its n_1-endpoint image is compact, because (i_1,s)\mapsto nS_{rw_q}(i_1\exp_N\sigma(s)) is continuous. No continuity of the anchor assignment or generation selectors is required; the tile partition is Borel and its anchor is fixed on each tile.

The bridge and the two target restrictions. We now estimate the bridge in (8.10). At the pivot, conjugation by d_q^{-1} cancels the character scaling S_{rw_q} and leaves only polynomially bounded unipotent factors. As s ranges in a fixed cube, the resulting logarithm has norm at most e^{C_1\delta r+o(r)}. Logarithmic distortion gives

where the constants are independent of sufficiently small fixed \delta. We retain this error at its stated size.

For good probes put

The first probe path has target relative heights in r(AT_1+o(1)), because its profiles approach A\pi z+b_1 and its pivot height is w_q. Its length is O(r). Path lifting and packing therefore give a lift of y_0^{-1}y_1 with N-part in B_r(AT_1,e^{o(r)}) and D-length O(r). Its endpoint height is o(r), since both endpoint profiles approach Aw_q+b_1. Every pure quotient width is zero, so

Choose its envelope large enough that the same restriction holds after fixed bounded right thickening of the endpoints. Indeed a bounded final step keeps the first path’s relative heights in r(AT_1+o(1)): its unthickened endpoint height is o(r) and 0\in AT_1. The same path lifting estimate applies.

For a tile contributing a good probe, the second starting path and the profile conditioning give

The path from y_{2,\kappa} to y_2 has relative heights in r(AT_2+o(1)), length O(r), and endpoint height o(r). Choose its lift u_2d_2 with

A lift of the bridge from y_2 to y_1, using (8.11) and the quasi-isometry bounds, has N log norm at most e^{C_3\delta r+o(r)} and D-length O(\delta r)+o(r). Conjugating its N-part by d_2 changes this estimate only by a subexponential factor. We obtain a lift of y_{2,\kappa}^{-1}y_1 of the form

The last height bound is only O(\delta r), not o(r).

Apply the absorption part of Lemma 8.2 to the target ideal I'=I_{S'} and template AT_2. After decreasing \delta, the ideal coordinate of u_2e_r belongs to B^{I'}_r(AT_2,e^{o(r)}). More explicitly, factor u_2=i_2\exp\sigma'(s_2), with \|s_2\|\le e^{o(r)}. Factoring \exp\sigma'(s_2)e_r again produces an ideal error with log norm e^{O(\delta)r+o(r)}. Every mixed ideal width is strictly positive, so this error fits in the ideal box. This is an ideal-coordinate bound uniform over the accompanying quotient coordinates; it does not require their location to stay in a subexponential ball.

The bound remains valid under fixed bounded right thickening of the lifts in (8.14). Such thickening adds an N-factor conjugated by d. The stated D-length and O(\delta r)+o(r) height bounds make its logarithm at most e^{O(\delta)r+o(r)}, and the same positive ideal widths absorb it. This explicitly accounts for the non-sublinear height of the lift relative to the second-sheet anchor.

All constants in the bridge and polynomial coordinate estimates can be chosen independently of sufficiently small \delta: the radii are fixed, the pivots stay in fixed compact height sets, and the quasi-isometry constants are fixed. There are finitely many cones, letters, and labels. We choose \delta_2 small enough to satisfy all their strict gap and absorption inequalities simultaneously.

The fixed-\delta bridge has now been absorbed in the ideal-coordinate bound. It remains to compare quotient Haar volumes: their exact multiplier will detect \ell_{S'}(b_2-b_1).

Volume at an arbitrary translated quotient location. Fix a contributing tile and let g_\kappa=y_{2,\kappa}^{-1}y_0. Equivariance in Lemma 8.1, together with (8.12), gives the exact restriction

It also holds for the thickened endpoints described above. The Haar multiplier and (8.13) imply

The set may be far from the quotient identity. Its location has no effect on this equality of Haar multipliers.

In the ideal and section coordinates (8.4), every admissible N-part of the thickened lifts lies in a set with ideal coordinate in B^{I'}_r(AT_2,e^{o(r)}) and quotient coordinate in g_\kappa\cdot \mathcal T_r. Quotient integration and (8.3) bound its Haar volume by

This is a bound by a containing product set. It requires neither independence of the coordinates nor a product description of the admissible set, and holds at any translated quotient location.

To turn this into a separated-point bound, take image endpoints in this tile with a fixed sufficiently large separation. Choose the admissible lifts relative to y_{2,\kappa} and thicken each on the right by a fixed small neighborhood in N\rtimes D. Their projections lie in disjoint metric balls, so these lifted neighborhoods are disjoint even if N\rtimes D\to G is not injective. Each has the same positive product Haar volume. The thickened D-coordinates lie in an intrinsic D-ball of radius O(r), of polynomial volume. The two coordinate restrictions just proved still hold for all the neighborhoods. Fubini’s theorem and (8.17) therefore bound their number by

The envelopes here are common to all contributing tiles, since the probe paths, profile tolerances, and coordinate bounds were uniform over their parameters.

The good n_1-volume was at least a constant times e^{rP_1}. Lemma 3.8, at the common pivot d_q, supplies at least e^{rP_1-o(r)} suitably separated source points from that set. Take the separation large enough that their images are separated at the constant used in (8.18). Assign each chosen point to its probe tile. Summing (8.18) over at most e^{r(P_1-P_2)+o(r)} tiles gives

Divide logarithms by r and let r\to\infty. We have proved \ell_{S'}(b_2-b_1)\le0.

For the reverse inequality take the original starting suffix to be k_0k and the step to be k^{-1}. The former is still in the allowed enlarged box by normalized multiplication, with an enlarged common subexponential envelope; the latter has the same generator amplitude bound. The swapped pair belongs to the swapped limiting support. Repeating the argument gives \ell_{S'}(b_1-b_2)\le0. Thus

The trace tests determine the offset. For the ray cone, its target trace is a positive multiple of \beta=(A^*)^{-1}\alpha. Hence \beta(b_2-b_1)=0. For each generator weight \lambda independent of \alpha, the target two-generator trace is c\beta+d\eta, where \eta=(A^*)^{-1}\lambda and c,d>0. Equation (8.19) then gives \eta(b_2-b_1)=0. Weights proportional to \alpha require no additional test. Since the generator weights span W^*, their target transforms do as well, proving b_2=b_1. When \dim W=1, the ray test alone gives this conclusion.

Finally, compactness converts the support assertion into convergence in probability on each compact subset: any violation would admit a joint limit assigning positive mass to unequal restrictions. The same contradiction works along arbitrary sequences of maps, basepoints, and allowed deterministic choices with their fixed common envelope. Conditioning on independently sampled suffixes and steps, or selecting violating deterministic choices if convergence failed, gives the stated random-choice version.□

Proof. By Lemma 3.3, an allowed suffix is a product of a bounded number of generator exponentials. Their amplitudes obey the bounds of Proposition 8.3; all partial products remain allowed suffixes by Lemma 3.1. The joint law of each successive pair of profiles is relatively compact by Proposition 6.7. Every limit is supported on identical pairs by Proposition 8.3. The distance between the two profiles on Z therefore tends to zero in probability. Summing over the bounded number of steps proves (9.1). If independent random suffixes violated the assertion, conditioning would select deterministic suffixes with failure probability bounded below. This would contradict the deterministic statement.

Write \mu_r for uniform Haar probability on B_r(\mathcal C). The normalized product bounds give a fixed L\ge1 such that

Let k be independent and uniform on B_r(\mathcal C,L). For each fixed a, the conditional law of ak dominates c\mu_r for a constant c>0 independent of r and a. Indeed, its Haar density on aB_r(\mathcal C,L) is the reciprocal of the volume of that box, and the ratio of this volume to \vol B_r(\mathcal C) is bounded. Consequently, for independent a,b with law \mu_r,

The ordinary profile laws are relatively compact and uniformly bounded on Z. If \nu is any subsequential limit, the last display says that \nu\times\nu is supported on the diagonal. Thus \nu is a point mass. Proposition 6.7 says that this point has the form A\pi+b, with A\in\mathcal A.

Set

and choose A_{F,x,r}\in\mathcal A to minimize the expected supremum norm in (9.2). These quantities are well defined because the profiles are bounded. The point-mass conclusion proves convergence to zero along every sequence of maps and basepoints; otherwise a violating sequence would have a point-mass subsequential limit for which the minimum is zero. This proves the stated supremum over F,x. Finally |a|_G\le Cr by Lemma 3.4, so the Lipschitz bound for F gives |s_{F,x,r}|\le Cr. Equation (9.1) transfers the deterministic approximation to the allowed suffixes.□

Proof. Put q=1+\delta/2. If r\le r'\le qr, let a be uniform on B_r(\mathcal C) and let k be independent and uniform on B_{r'}(\mathcal C,L), for a sufficiently large fixed L. For every such a,

The density argument in Lemma 9.1 shows that the law of ak dominates a fixed multiple of uniform measure on B_{r'}(\mathcal C). This suffix fits the strict expansion at scale r, since r'\le(1+\delta/2)r.

For the same N-coordinate n, compare the two deterministic approximations at

On a fixed smaller Z, z' stays in the original neighborhood. The errors of both approximations are o(r) in probability, uniformly over r'\in[r,qr], F, and x. To justify this last uniformity, a violating sequence would be an allowed sequence in Lemma 9.1, with the fixed envelope coming from L; the density comparison constants are fixed. More precisely, the lower domination of the law of ak transfers its scale-r good event to uniform measure on B_{r'}(\mathcal C). Intersect there with the ordinary scale-r' good event. Therefore

The expression is deterministic: an event of positive probability on which both approximations are accurate proves its bound.

Subtract (9.3) at 1 from its values at a fixed finite set of points whose heights span W. The offsets cancel. Distinct members of the finite set \mathcal A are separated on these heights, so the labels agree for all r\ge R_0, with one R_0. Since any larger scale can be reached by successive ratios in [1,q], the label stabilizes. Evaluating at 1 also gives

uniformly in the other variables.

For completeness, fix \epsilon>0 and choose a fixed R\ge R_0 such that \varepsilon(t)\le\epsilon for t\ge R. Telescope along scales R,qR,q^2R,\ldots, ending with a ratio at most q. The uniform bound at the initial scale gives

First divide by r and let r\to\infty, then let \epsilon\to0. This proves the uniform sublinear bound.□

Proof. It remains to compare the stabilized labels and center values at different basepoints. Fix a sufficiently small \eta>0 relative to \delta. Suppose |x^{-1}y|_G\le\eta r and choose a lift

using Lemma 3.6. Set r_0=(1-\delta/2)r and, for a uniform on B_r(\mathcal C), put

Its law has constant Haar density 1/\vol B_r(\mathcal C) on its support, since conjugation by d has determinant one.

Let a_0 be uniform on B_{r_0}(\mathcal C) and let k be independent and uniform on d^{-1}B_r(\mathcal C,L)d. We claim that the law of a_y is dominated by a fixed multiple of the law of a_0k. Indeed, for every possible a,a_0,

The inclusion follows from Lemma 3.1: the smaller box retains a strict exponent gap after conjugation by d, and u fits strictly within the r-widths if \eta is small enough. Polynomial conjugation factors are absorbed by these gaps. The Haar volume of the box for k is a fixed multiple of \vol B_r(\mathcal C). Conditional density comparison, followed by integration in a_0, proves the claim.

All these k lie in the allowed expansion at scale r_0. For this use (1+\delta)(1-\delta/2)>1 and choose \eta smaller if necessary; the remaining strict gap again absorbs the polynomial factors. Thus Lemma 9.1 applies with a common envelope to this collection of comparisons.

We have the exact identity

For z in a fixed smaller neighborhood, the point z'=\Delta_{r_0}^{-1}(d^{-1}\Delta_rz) lies in Z. This follows from Lemma 2.2 and the small bound on |d|_D/r; its height satisfies exactly

The two deterministic chart approximations, transferred through the fixed density domination, can now be compared. Their offsets are o(r) by Lemma 9.3. Subtracting the comparison at two values of z cancels both center values and \pi d. Testing a fixed spanning set of heights forces equality of the stabilized labels at x,y, uniformly for all sufficiently large r. Evaluation at z=1 then gives

uniformly in F,x,y.

Any two basepoints can be compared at a sufficiently large scale; therefore the stabilized label is a single A_F. To obtain (9.4) for |g|\le R, take y=xg and r=R/\eta. The preceding uniform error then provides one modulus \epsilon(R)\to0 for all maps and basepoints.

Finally apply (9.4) to both maps in a coarse-inverse pair. For g_t=\exp_D(tH), the displacement F(g_t) from F(1) is O(t). Bounded-error composition gives

Every w\in W is \pi H for some H\in\mathfrak d. Division by t proves A_{\bar F}A_F=\id.□

Proof. Since \mathcal A^* permutes the generator rays, the source quotient is defined. Choose v\in\mathfrak d with \alpha(\pi v)<0. There are only finitely many ray and label choices, so choose all these v’s once.

Convergence and a uniform compact shadow. For x_t=x\exp_D(tv), Proposition 9.4 gives

with c_0,T_0>0 uniform in F,x and the finite choices. Set y_t=F(x)^{-1}F(x_t). Its distance from 1 is O(1+t). By Lemma 3.6 it has a lift y_t=n_td_t with |d_t|_D\le C(1+t). The increments y_t^{-1}y_{t+1} have uniformly bounded length and hence bounded lifts. Equations (10.2) and (10.3) imply

The series is summable. In a left invariant Riemannian metric, d(1,\exp Z)\le C\|Z\|, by using the path s\mapsto\exp(sZ). Thus the projected trajectory at integer times is Cauchy and stays in a fixed closed ball. Completeness and properness of the metric give convergence in a fixed compact set. The same estimate for time increments of size at most one gives convergence for real t\to\infty. The initial interval 0\le t\le T_0 contributes a fixed compact set, so the limit lies in a fixed compact Q_\beta. Equivariance of p_\beta under the G-action therefore gives a limit

Independence of the representative. We first record the path estimate used for this purpose. Suppose, for large t, a source path stays at distance at most C_2t from a fixed x, has length bounded by a polynomial in t, and satisfies

at each of its points z. The uniform sublinear estimate puts its image, relative to F(x), at \beta-heights at most -c_2t/2 for all sufficiently large t. The lifts at all image points have D-length O(t), because their distance from F(x) is O(t). Subdivide the source path into bounded-length pieces. Equations (10.2) and (10.3) bound each projected image increment by a polynomial times e^{-ct}. There are only polynomially many pieces, so the projected distance between the endpoints tends to zero. Constants here may depend on this fixed collection of paths.

For fixed d\in D, connect x\exp(tv) to xd\exp(tv) by xd(s)\exp(tv), where d(s) is a fixed piecewise smooth path in D from 1 to d. Its length is polynomial in t, since conjugation in the nilpotent group D has polynomial norm. Its points have distance O(t) from x, and their relative \alpha-heights equal t\alpha(\pi v)+O(1). The preceding path estimate gives b(xd)=b(x).

For fixed u\in I_\alpha, we will vary the height within the halfspace \alpha<0 to realize u while keeping the \alpha-height strongly negative. Choose a compact convex set E\subset W containing \pi v, entirely in \alpha<0, such that every bracket word containing an outside-ray letter has h_Z(E)>0. Here is why such an E exists. Every generator form outside the positive \alpha-ray is unbounded above on the halfspace \{\alpha\le\alpha(\pi v)\}. For each outside form choose a point in that halfspace on which it is as large as desired, and take the convex hull together with \pi v. The support maxima for inside-ray letters remain fixed negative numbers. Since the list of words and their lengths is finite, sufficiently large outside maxima make the support sum of every mixed word positive.

By Lemma 8.1, the mixed words span \Lie(I_\alpha). Thus the fixed element u belongs to B_t(E,O(1)) for all sufficiently large t. One can see this directly in the all-word description of the box: the fixed coordinates of \log u in a mixed-word basis are dominated by its positive exponential widths. Choose an affine section \sigma:W\to V of \pi with \sigma(\pi v) equal to the abelianization of v. The compact convex set E_V=\sigma(E) contains that abelianization and satisfies \pi E_V=E. The paths of Lemma 3.5, applied to E_V and the pivot \exp(tv), connect \exp(tv) to u\exp(tv) in length O(t) with heights in t(E+o(1)). All their heights remain strictly negative in \alpha by a linear amount, and all their points have distance O(t) from 1. Left translation by x and the preceding path estimate give b(xu)=b(x).

The stabilizer of 1\in L_\alpha is I_\alpha D. The two invariances therefore show that b(x) depends only on p_\alpha x. Define f_\beta(p_\alpha x)=b(x). This establishes (10.4).

Inverse and continuity. By Proposition 9.4, A_{\bar F}=A^{-1}, so the construction for \bar F gives \bar f:L_\beta\to L_\alpha. For x_t=x\exp(tv) the quotient point p_\alpha x_t is fixed. Equation (10.4) at x_t supplies representatives z_t of f_\beta(p_\alpha x) within a fixed distance of F(x_t): lift the compact set Q_\beta to a bounded set in G. The coarse-inverse estimate and the shadow estimate for \bar f then give

for a fixed compact Q'\subset L_\alpha. The sets on the right contract to p_\alpha x, since \alpha(\pi v)<0. Thus \bar f f_\beta=\id. Interchanging the maps proves f_\beta\bar f=\id.

Fix x and t. The projection of a fixed-radius neighborhood of x_t contains a neighborhood of p_\alpha x; this uses only that the quotient projection is a submersion. By the Lipschitz bound and (10.4), its image under f_\beta is contained in F(x_t)\cdot Q'', with Q'' fixed. These sets have diameter tending to zero. Indeed, after normalization by F(x), the D-part has length O(t) and negative linear \beta-height, so (10.3) applies; quotient left translations are isometries. The fixed action of F(x) is Lipschitz, being a left translation followed by an automorphism. This proves continuity at p_\alpha x. The same argument proves continuity of \bar f, completing the proof.□

Proof. Fix a unit ball V_-\subset L_\beta and a bounded set of lifts of radius R_- in G. Denote the common Lipschitz constants by L_F,L_{\bar F} and the common coarse-inverse error by E_0. For each such lift b,

The shadow estimate for the inverse boundary map therefore puts f_\beta^{-1}(F(x)\cdot V_-) inside x\cdot U, where U is one compact ball containing

Here Q_{\rm inv} is its fixed shadow compact. This choice of U is independent of x.

Choose a bounded lift set for this U once, of radius R_U. For every lift b in it, d_G(F(xb),F(x))\le L_FR_U. The forward shadow estimate gives f_\beta(x\cdot U)\subset F(x)\cdot V_+ for one compact V_+. Thus

Both comparison sets have positive finite Haar volume. The G-action scales that volume by e^{\ell_\beta(\pi F(x))}, proving the assertion. There are finitely many ray and label choices, so all constants can be chosen uniformly.□

Proof. Fix a generator X_i and any t\in\R. For every source generator ray other than that of \alpha_i, the element \exp(tX_i) lies in its ideal and acts trivially on the corresponding quotient. Hence the two source sets

are identical. Lemma 10.2 bounds the difference of the associated target trace heights by a uniform constant. By (10.1), this bounds every target ray-height difference except possibly the ray paired with \alpha_i.

There is a relation \sum_\rho c_\rho\rho=0 over all distinct target generator rays, with every c_\rho>0, by Lemma 3.2. Applying it to the height difference bounds the one remaining form as well. Since the ray forms span W^*, there is a uniform constant B_1 with

for every x,i,t. No bound on t is imposed.

The sets B_r(\mathcal C) exhaust N, since every generator support maximum on \mathcal C is positive. By Lemma 3.3, every such box element is a product of at most one fixed number M of generator exponentials. Their parameters can be arbitrary, because the preceding estimate is parameter independent. Thus B_N=MB_1 works for every n. Finally nd=d(d^{-1}nd) and d^{-1}nd\in N; applying the bound at the basepoint d proves the last assertion without introducing any dependence on d.□

Proof. For compact E\subset B(R_E) and T>R_E, Tonelli’s theorem gives

Nilpotent groups are unimodular, so inversion preserves Haar measure. The inner quantity equals \mu(E) for y\in B(T-R_E), vanishes outside B(T+R_E), and lies between zero and \mu(E) everywhere. Consequently

Fix a large-scale doubling constant D_0 such that \mu(B(2T))\le D_0\mu(B(T)) for all sufficiently large T. Such a constant follows from the polynomial volume estimates for nilpotent groups, as in Lemma 2.2. Put q_0=a_0/\mu(E_0). Apply (10.7) to E_0, first with averaging radius T-R_{E_0} and then with radius T+R_{E_0}. The assumed translate bounds and doubling give, for sufficiently large T,

For a fixed j, choose T so large that these bounds apply to T-R_{E_j} and T+R_{E_j} and that T\ge2R_{E_j}. Use (10.7) for E_j and the assumed translate bounds once more. Doubling yields

The required averaging radius may depend on j; the displayed constants do not. Only integration of nonnegative measurable functions was used, so the proof applies equally to singular Radon measures.□

Proof of Theorem 1.4. If W=0, the assertion is immediate. Otherwise first take a Lipschitz pair from the uniform family of Lemma 5.1, with label A_F from Proposition 9.4. Fix a target ray \beta and its source ray \alpha=\beta\circ A_F. Let f_\beta and U be given by Lemmas 10.1 and 10.2. Define a Radon measure on L_\alpha by

This is a measure because f_\beta is a homeomorphism; no absolute-continuity property is asserted.

For d\in D, put U_d=d\cdot U and a_d=e^{\ell_\beta(\pi F(d))}. Every z\in L_\alpha has a representative n\in N, and zU_d=nd\cdot U. Lemmas 10.2 and 10.3 give

with one uniform comparison constant. Lemma 10.4, applied to these sets and the reference set U_1=U, implies

The source action scales Haar by e^{\ell_\alpha(\pi d)}. Taking logarithms therefore gives

with a uniform error.

We identify its linear term using the already proved sublinear estimate. For w\in W, choose H\in\mathfrak d with \pi H=w and set d_t=\exp_D(tH). Its length is O(t), so (10.8) and Proposition 9.4 give

After division by t, \ell_\alpha=\ell_\beta\circ A_F. The target ray traces span W^* by (10.1). A fixed spanning subfamily of (10.8) therefore yields

with uniform B_D. For any g=nd, Lemma 10.3 and \pi n=0 extend this bound to g, with bound B_D+B_N. All finite ray and label choices were absorbed into the constants, so the conclusion holds for the whole fixed family.

Finally an original (K,C) quasi-isometry is at a uniformly bounded distance from its Lipschitz approximation by Lemma 5.1. Since \pi is Lipschitz, changing back alters the error by a uniform constant, proving the stated theorem for the original maps.□

Proof. Dekimpe’s lattice realization theorem [5, Theorem 4.1] gives \Gamma and S, with \Gamma normal of finite index in P if desired. The uniform lattice acts properly and cocompactly by left translations on S. Thus its inclusion is a quasi-isometry by the Milnor–Švarc lemma; the same observation applies to a finite-index subgroup acting on a Cayley graph. See [2, Theorem 4.C.5(1) and Proposition 4.C.11(3)]. Cornulier–Tessera’s reduction [3, Lemma 3.A.1] gives a real-triangulable G quasi-isometric to S. It remains to check unimodularity and the stated properties of H and \Gamma.

The group P is amenable, since cyclic groups are amenable and this property is preserved by extensions and finite extensions. We give the metric Følner argument that transfers this information to G and H. It is important here to obtain metric Følner sets in G, rather than only amenability as a locally compact group; compare [28, Proposition 11.11 and Corollary 11.13].

Let Y denote either G, with a left invariant Riemannian metric and left Haar measure \mu, or H, with a word metric and counting measure. Choose a quasi-isometry u:P\to Y with coarse-surjectivity radius c>0, and fix a>c. For finite nonempty E\subset P, put

For each fixed radius, the lower quasi-isometry inequality bounds the number of points of P whose images lie in a ball of that radius. Consequently the balls in this union have bounded multiplicity, and

for a constant c_1>0 independent of E.

Write N_R(A) for the metric R-neighborhood of a set A. If y\in N_R(A_E)\setminus A_E, choose p_y\in P with d_Y(y,u(p_y))\leq c and p\in E with d_Y(y,u(p))\leq R+a+1. Since a>c, we have p_y\notin E. The lower quasi-isometry inequality gives d_P(p_y,E)\leq R', with R' depending only on R,a and the quasi-isometry constants. It follows that

An increasing-radius Følner sequence in P, together with (11.1)–(11.2), therefore gives finite-measure sets A_j\subset Y satisfying

For the discrete group H, this is the Følner criterion for amenability.

For Y=G, right translation by g moves every point a distance d_G(1,g), so A_jg\subset N_{d_G(1,g)}(A_j). If right translation multiplied left Haar measure by a factor greater than one, this inclusion would contradict (11.3). A nontrivial modular homomorphism has such a value, after replacing g by g^{-1} if necessary. Thus G is unimodular.

Finally, a connected simply connected solvable Lie group of dimension n is diffeomorphic to \R^n; explicit coordinates are given in [3, Lemma 3.B.2]. The compact smooth manifold \Gamma\backslash S is therefore aspherical and has a finite CW structure. It is a finite classifying space for \Gamma, and manifold Poincaré duality makes \Gamma an integral Poincaré duality group of dimension n.□

Proof. Fix a finitely generated subgroup J\leq K_0 and a finite symmetric generating set T of J. The heights of all orbit points f(j), j\in J, have norm at most D. For a word j=t_1\cdots t_m with m\leq t, put j_i=t_1\cdots t_i. The upper quasi-isometry inequality gives a constant L_T such that

Join these successive points by minimizing geodesics. The resulting path starts at 1, ends at f(j), and has length at most L_Tt. The homomorphism \pi is Lipschitz for a left invariant Riemannian metric: its differential has constant norm in left-trivialized tangent spaces. If L_\pi is a Lipschitz constant, every point of the path has height norm at most D+L_\pi L_T.

Lemma 3.7, with these fixed length and height bounds, shows that a maximal 1-separated subset of f(B_T(t)) has at most C_T(1+t)^{b_T} points. The unit balls about that subset cover f(B_T(t)). The lower quasi-isometry inequality bounds the number of elements of H mapping into each such ball by one constant: any two such elements are at uniformly bounded distance in the fixed word metric of H. Hence

This is an intrinsic growth bound for J; no undistortion of J in H is assumed. Gromov’s polynomial-growth theorem [18, Main Theorem, p. 54] makes J virtually nilpotent.□

Proof. Choose quasi-inverse maps f:H\to G and r:G\to H. Bounded changes allow us to arrange f(e)=1 and r(1)=e. For h\in H, define

The maps F_h have uniform quasi-isometry constants, because the maps L_h are isometries. There is a constant D_0 such that

and F_e is uniformly close to the identity. These assertions follow from the bounded errors in r\circ f and f\circ r. Also F_h(1)=f(h).

Put s(h)=\pi f(h). The assumed height theorem gives a single B<\infty and matrices A_h in the fixed finite group \A_G such that

Apply this formula twice to F_hF_k(g), and compare with its application to F_{hk}(g) using (11.5). It follows that

is bounded as g ranges over G. Since \pi is surjective onto the vector space W, its linear coefficient vanishes. Similarly A_e=I. Thus h\mapsto A_h is a homomorphism, and

is normal and has finite index. Evaluating the same comparison at g=1 gives a constant D_1 such that

The uniform constant in the height theorem is essential here: D_1 does not depend on h or k.

The subgroup H_0 is amenable. Let m be a left-invariant mean on bounded real functions on H_0, extended coordinatewise to functions with values in W. Define

For fixed h, the function being averaged is bounded, since (11.7) bounds its distance from s(h) by D_1. Positivity of the mean gives

To verify additivity, write

Left invariance under x\mapsto kx shows that the means of the two terms on the right are \tau(h) and \tau(k). Therefore \tau(hk)=\tau(h)+\tau(k).

Let K_0=\ker\tau. Equation (11.9) is the hypothesis of Lemma 11.2; that lemma makes each finitely generated subgroup of K_0 virtually nilpotent. Nilpotent groups are elementary amenable by their central series, and so are their finite extensions. The finitely generated subgroups of K_0 form a directed family with union K_0. Thus K_0 is elementary amenable. The exact sequence

and abelianness of \tau(H_0) show that H_0 is elementary amenable. Finally H/H_0 is finite, so H is elementary amenable as well.□

Proof of Theorem 1.1. Let P and H satisfy the hypotheses of the theorem. Apply Lemma 11.1 and retain both the real-triangulable unimodular model G and the uniform lattice \Gamma\leq S.

If W=0, Lemma 2.3 makes G nilpotent. Its metric balls have polynomial Haar volume, by the nilpotent coordinate bounds of Lemma 2.2. A quasi-isometry from H to G has bounded multiplicity on bounded scales, so disjoint small-ball packing transfers this to polynomial word growth of H. Gromov’s theorem [18, Main Theorem, p. 54] makes H virtually nilpotent, hence virtually polycyclic. This also includes the zero-dimensional model: in that case H is finite. We may therefore suppose W\ne0.

By Lemma 11.1, H is amenable. Apply Theorem 1.4 to the uniform family transferred from its left translations. Proposition 11.3 shows that H is elementary amenable.

The group \Gamma has a finite classifying space and is quasi-isometric to H. Li’s coarse invariance theorem [23, Corollary 4.47], applied over \Z for each n, gives H\in\FP_\infty(\Z). The elementary amenable finiteness theorem [21, Theorem 1.1, (viii)\Rightarrow(v)] now gives a finite-index subgroup V\leq H of type F. The structural conclusion recorded in Section 2 of the same paper also gives a finite-index solvable subgroup U\leq H. Set H_1=U\cap V. Then H_1 is solvable and has finite index in H. The finite cover of a finite classifying space for V corresponding to H_1 is a finite classifying space for H_1. In particular,

This also implies that H_1 is torsion-free: restriction of a finite projective resolution to a hypothetical nontrivial finite cyclic subgroup would give finite integral cohomological dimension for that subgroup, which is impossible.

Both H_1 and \Gamma now have finite integral cohomological dimension, and they are quasi-isometric. Since \Gamma is an integral Poincaré duality group, Li’s duality transfer [23, Corollary 4.48], with coefficient ring \Z, makes H_1 an integral Poincaré duality group. Here the type-F subgroup supplies the finite-cohomological-dimension hypothesis. Bieri’s theorem [1, Theorem 9.23] says that a solvable integral Poincaré duality group is polycyclic. Hence H_1 is polycyclic, and H is virtually polycyclic.□

Proof of Corollary 1.2. Let \Lambda be a lattice in a connected simply connected solvable Lie group S. Such a lattice is polycyclic [13, Proposition 3.4], and is uniform [15, Theorem 2.9(i)]. It is therefore finitely generated and quasi-isometric to S by the Milnor–Švarc lemma. A finitely generated group H quasi-isometric to \Lambda, or equivalently to S, is virtually polycyclic by Theorem 1.1. Dekimpe’s realization theorem [5, Theorem 4.1] then embeds a finite-index subgroup of H as a uniform lattice in some connected simply connected solvable Lie group.

Conversely, every virtually polycyclic comparison group has such a finite-index lattice model by the same realization theorem. Finite index preserves quasi-isometry, so the lattice formulation implies the group formulation. The ambient Lie group obtained for H is not prescribed by this equivalence.□

Proof of Lemma 5.1. A coarse inverse can first be chosen with constants depending only on K,C. Indeed, for each target point choose a source point whose image is within the coarse-surjectivity constant. The two quasi-isometry inequalities bound the distortion of this choice and its two composition errors. It is therefore enough to smooth either member of a uniformly controlled coarse-inverse pair by a bounded amount.

Choose a maximal 1-separated set \mathcal V\subset G. Properness makes it locally finite, and maximality makes its closed unit balls cover G. The cover by open balls of radius 2 has uniformly bounded multiplicity: disjoint balls of radius 1/3 about participating centers lie in a ball of fixed radius, and left invariance gives the same Haar-volume ratio at every center. Its nerve therefore has bounded dimension, say q; realize each simplex as a standard regular Euclidean simplex with unit edges. The functions

define a partition map \lambda from G to the nerve. The denominator is at least 1. If M bounds the cover multiplicity, normalization and the 1-Lipschitz cutoffs give \|\lambda(g)-\lambda(h)\|_1\le4M d(g,h). This also controls the intrinsic piecewise Euclidean metric. Indeed, for probability vectors a,b with \|a-b\|_1<2, normalize their coordinatewise minimum to a probability vector c. The vector c lies in the common face of their support simplices, and the two segments a to c and c to b have total Euclidean length at most 2\|a-b\|_1. Thus \lambda is locally 8M-Lipschitz for the intrinsic metric. Subdivision of geodesics in G makes this a global bound.

Let J be either coarse map. Map the vertex v to J(v). The image vertices of a simplex have uniformly bounded diameter, since its source vertices have pairwise distance at most 4. Extend over simplices by increasing dimension, always preserving the maps already defined on their faces. The required controlled extension uses exponential coordinates on G, which are global by Lemma 2.1. For a center p\in G put

For every fixed R, on d(p,z)\le R this contraction has a uniformly bounded derivative in s and spatial Lipschitz constant at most c_Rs. To see the latter bound, differentiate the smooth map \exp(s\log z) on a compact set. Its spatial derivative vanishes at s=0, so the mean-value estimate in s supplies the factor s. For the spatial distance estimate, bound derivatives on an enlarged compact ball containing geodesics between points of the original ball. Left translation makes the bounds independent of p.

On a simplex write a point as b+s(u-b), where b is its barycenter and u lies on its boundary. If H is the previously constructed boundary map, extend it by \mathcal C_p(s,H(u)), taking p to be the image of one vertex. Facewise Lipschitz bounds give a Lipschitz boundary map for the Euclidean distance, since the boundary of a regular simplex of dimension at least two has bounded intrinsic quasiconvexity constant. For an edge its two endpoint images already have bounded distance. The radial extension is quantitatively Lipschitz: if \rho is the distance from b to the boundary and D is the maximum distance from b to the simplex, then for x=b+s(u-b) and x'=b+s'(u'-b) one has

Combine these with the spatial O(s) bound and the uniform time derivative bound of \mathcal C_p. Regular simplices of bounded dimension have uniform values of the geometric constants involved. The extension also remains in a ball of controlled radius about the vertex images, because the contraction is evaluated on a fixed compact ball. Induction through at most q dimensions gives one uniform Lipschitz constant and one uniform radius. Shared faces retain their previous maps, so the extensions agree. Simplexwise Lipschitz bounds give a Lipschitz map for the intrinsic metric of the nerve.

Compose this map with \lambda. Whenever \lambda_v(g)>0, one has d(g,v)<2, hence d(J(g),J(v))\le 2K'+C' for the uniform constants of the coarse map under consideration. The controlled radius of the simplex extension therefore bounds the distance of the resulting Lipschitz map from J, uniformly in g.

Apply this construction to both coarse maps. Bounded perturbations preserve the lower quasi-isometry inequalities. They preserve the bounded composition errors as well: use the original coarse upper inequality to compare the original map at two points a bounded distance apart, and then the bounded perturbation of its values. Enlarging a single constant \Lambda gives all the stated bounds.□