A Modulus Proof of Cannon’s Conjecture

Proof. Base change and thin triangles give

Thus a^{-(v|y)_e}\asymp D_y. Exponentiating the first identity and using (hy|hz)_e=(y|z)_v+O(1) proves (2.2). With z=z_0, the factor D_{z_0} is bounded above and below, and d(y,z_0) is bounded above. When d(x,y)\le\varepsilon_*/2, the latter is also bounded below. This proves (2.3). Finally, (v^{-1}|v^{-1}z_0)_e=k-(v|z_0)_e+O(1), and (v|z_0)_e is bounded in terms of \varepsilon_*.□

Proof. Translate a bi-infinite geodesic so that it passes through v. The ideal triangle with third vertex e puts v uniformly near one of its based sides; the nearby parameter differs from |v| by a bounded amount. A bounded change of basepoint changes products by a bounded amount, preserving (2.2). The two remaining assertions follow from (2.1) and the product estimate.□

Proof. Choose k with r_k\asymp t. Rays ending in the given ball have their depth-k vertices in a uniformly bounded graph ball, by Lemma 2. At depth k+j there are at most CA^j possible continuations, where A>1 depends on the valence. Endpoints whose rays share such a vertex are at visual distance at most C'r_{k+j}. Choose the least j\ge0 with C'r_{k+j}<s. The desired bound follows with Q\ge\max\{2,\log_a A\}. Coarse scales are absorbed by increasing the constant.□

Proof. For small s, expand at x with R\asymp s and choose z_0 a uniformly positive distance from x; two fixed distinct points suffice for this choice. By (2.3), the two image points stay a uniform positive distance from p=hz_0. Fix a homeomorphism \phi:Z\to\Sphere. Uniform continuity of \phi^{-1} places their spherical images a uniform distance from \phi(p). Join them outside a smaller round ball about \phi(p). Uniform continuity of \phi makes the pullback path stay a uniform d-distance from p. The upper bound in (2.3) puts its preimage under h in B(x,Ls). Increase L for the remaining scales, using path connectedness of Z.

For a continuum C in an open set V, choose \varepsilon>0 with its L\varepsilon-neighborhood in V. Connectedness supplies a finite \varepsilon-chain in C between any two specified points: the set reachable by such chains is relatively clopen. Join consecutive points by the first assertion.□

Verification of the cited criterion. Corollary 3.5 of [8], derived from [6, Theorem 10.4], applies to an arcwise connected, approximately self-similar metric sphere. Approximate self-similarity means that each open ball, with distances divided by its radius, is uniformly bi-Lipschitz to an open subset of the space. Lemma 1 supplies this property; intermediate and large radii change only the constant.

The criterion assumes bounded 2-moduli of paths of diameter at least a sufficiently small d_0 in successive graph approximations. For a dyadic scale 2^{-m} choose n with 2^{-m}/a<r_n\le2^{-m}. Our cells form a fixed \kappa-approximation for any \kappa>2a: the inner balls of radius 2^{-m}/\kappa are disjoint, each is contained in its cell, and each cell is contained in the radius-\kappa2^{-m} ball about its center. The modulus is exactly M(n). Finitely many coarse scales have finite modulus. No Coxeter-group assumption occurs in this criterion.□

Proof. Let \psi:Z\to\Sphere be the uniformizing homeomorphism and write \mathcal G=\{\psi g\psi^{-1}:g\in G\}. The boundary cross-ratio estimate and the two fixed maps \psi,\psi^{-1} give one quasi-Möbius distortion function for every element of \mathcal G. Fix f\in\mathcal G and x\in\Sphere, keep a fourth point away from x, and let two points at equal spherical distance from x approach x. In the cross-ratio inequality the factors involving the fourth point tend to one, both before and after applying f. Thus the infinitesimal linear distortion has a bound independent of f and x; the radius at which the estimate is accurate need not be uniform in f. The metric characterization of planar quasiconformality, applied in conformal charts, makes each f quasiconformal. At its almost-everywhere nonsingular differentiability points, the same infinitesimal bound controls the ratio of singular values of Df, and hence its analytic dilatation. This gives a common quasiconformal bound. For a reversing map apply the characterization after a conformal reflection; see [4, Theorems 2.5.3–2.5.4] and the metric characterization in [17, Theorem 1.4 and Remark 1.8(b)].

We recall explicitly the invariant-structure argument, including orientation reversal. This is the invariant-conformal-structure method of Sullivan, with the circumcenter construction detailed by Tukia [24, pp. 73–76]. Analytic quasiconformal maps have almost-everywhere defined nonsingular weak differentials, and they and their inverses preserve null sets [4, Theorem 3.1.2]. Their differentials satisfy the almost-everywhere composition rule [2, Lemma 10 and Theorem 5]. Since \mathcal G is countable, remove the \mathcal G-orbit of the union of all exceptional sets for these assertions and for all pairs of maps. Its complement X is invariant and conull. At x\in X, let \mathcal C_x be the space of positive definite quadratic forms on T_x\Sphere modulo positive scalars. This is a hyperbolic plane, and pullback by any invertible linear map, of either orientation, is an isometry between the corresponding fibers [4, Section 2.3, equations (2.17)–(2.18) and Theorem 2.3.2]. The orbit \{f^*q_{\rm round}(x):f\in\mathcal G\}\subset\mathcal C_x is uniformly bounded, by the common quasiconformal bound, and contains the round form. Its unique hyperbolic circumcenter defines a uniformly bounded measurable conformal structure q(x). Indeed, circumcenters of the increasing finite subsets from an enumeration are measurable and converge to that circumcenter: their radii increase, their centers stay in a fixed compact ball, and every subsequential limit is the unique center of the whole orbit. The composition rule permutes this orbit under pullback, so f^*q=q on X for every f\in\mathcal G. Extend q arbitrarily across the null complement of X.

The measurable Riemann mapping theorem [2, Theorem 6] gives a quasiconformal homeomorphism F:\Sphere\to\Sphere with F^*q_{\rm round}=q almost everywhere. Invariance and the analytic composition rule show that every FfF^{-1} is conformal or anticonformal, according to the orientation of f. Such sphere homeomorphisms are Möbius or anti-Möbius. Their unique extensions to hyperbolic isometries therefore define a homomorphism \rho:G\to\Isom(\HH^3).

For completeness, properness and cocompactness follow directly from the action on ordered distinct triples \mathcal T(Z). On any compact set of triples all pairwise distances are bounded below. For |v|\to\infty, at least two points of such a triple stay away from the expansion center for v^{-1}; their image distance is O(r_{|v|}). Thus v^{-1}K\cap K=\varnothing eventually for every compact K\subset\mathcal T(Z). Conversely, expand at one point of a closest pair with scale comparable to their distance. Formula (2.2) makes all three image distances uniformly positive: if the third point is nearby, all denominator factors have that scale; otherwise its distances to the pair cancel its denominator factor. Triples already uniformly separated need no expansion. Hence the action on triples is proper and cocompact.

These properties survive boundary conjugacy. Let I=\Isom(\HH^3). The stabilizer H of a standard ordered boundary triple has two elements: the identity and reflection in its hyperbolic plane. Choose o\in\HH^3 fixed by H, and let K_o be its compact stabilizer in I. The equivariant map

is continuous, onto, and proper: isometries sending o into a compact set form a compact set. Properness and cocompactness therefore descend. The kernel fixes o, so is finite.□

Proof. Cover Z by finitely many coordinate boxes [-1,1]^2, each lying in a chart containing [-8,8]^2 of diameter less than d_0. A macroscopic path meeting an inner box must leave its larger box. Follow it to its first meeting with \partial[-2,2]^2. If it exits through x=2, its segment after the last preceding meeting with x=1 crosses [1,2]\times[-2,2] from left to right. The other three exit sides give analogous strips. Each strip extends by its height in both transverse directions inside [-8,8]^2.

Consequently there are finitely many crossing families \mathcal F_1,\ldots,\mathcal F_J such that every macroscopic path contains a member of one of them. Summing admissible weights gives

One strip realizes this bound along a subsequence. Affine coordinates give the asserted rectangles.□

Proof. For (i), suppose a lateral continuum and a top-to-bottom path are disjoint. Enlarge the rectangle, extending the continuum horizontally and the path vertically to the corresponding new edges. The extensions remain disjoint: a new horizontal and vertical segment can meet only at an original corner, already a common point of the original sets. Approximate the extended path, within its positive clearance from the continuum, by a polygonal path with interior in the enlarged rectangle. After subdivision at intersections it contains a simple polygonal arc between the top and bottom edges. The Jordan curve theorem, applied after adjoining a boundary route, shows that this arc separates the lateral edges. The connected continuum cannot avoid it.

For (ii), subdivide into finer square grids and mark the closed squares meeting the blocker K. No edge-adjacent chain of unmarked squares joins top to bottom. Label A the unmarked squares reachable from the top through unmarked edge-adjacent squares, and append a top row of A squares. Label all remaining squares and an appended bottom row B. Draw the interfaces between unlike labels. Interior vertices have even degree; each lateral boundary has an odd number of ends, since its labels begin with A and end with B. There are no top or bottom ends. Some interface component therefore joins the lateral boundaries: otherwise every component contributes an even number of ends to the only lateral boundary it meets.

Along a path in this component, the adjacent B squares are original marked squares. An adjacent unmarked square would be reachable, while an interface with the appended bottom row would give an unmarked top-to-bottom chain. Successive B squares meet at least at a corner. Their union is a connected compact lateral crossing. A Hausdorff limit as the mesh tends to zero is connected, meets both lateral edges, and lies in K.□

Proof. Let P minimize Euclidean norm in the convex hull of the finitely many cell-incidence vectors v of left-to-right crossings. Its coordinates are nonnegative and sum to at least one, so P\ne0. Minimality gives \langle P,v\rangle\geq\|P\|^2, making P/\|P\|^2 admissible. Conversely any admissible \omega satisfies \langle\omega,P\rangle\geq1. Thus the left-to-right modulus is \|P\|^{-2}. By Lemma 6(i), each top-to-bottom incidence vector has inner product at least one with every v, hence with P. Set \rho=P and apply Lemma 5.□

Proof. We have F_N\geq U_N^0 and H_N\leq F_N; constant paths give the band values. Concatenation with nearly optimizing paths gives, for x,y\in K,

If F_N(x)>F_N(y), the first inequality places their values between \min_K U_N^0 and \max_K U_N^0. If H_N(x)>H_N(y), the second places them between \inf_K F_N and \sup_K F_N. This proves (3.3). Short coordinate segments and continuity of U_N^0 then prove continuity of both functions, including at the boundary.

Every point of \{H_N>t\} joins the bottom band within that set: choose a path with \inf F_N>t and use its tails. Likewise every point of \{F_N<t\} joins the top within \{F_N<t\}. To connect a point x with H_N(x)<t\leq F_N(x) to the top, choose H_N(x)<s<t and follow any top-going path until its first point with F_N=s. On this prefix F_N\geq s, so concatenation with nearly maximizing paths from any prefix point z gives

Hence H_N(z)\leq H_N(x)<t throughout the prefix. Its endpoint joins the top through F_N<t, hence through H_N<t. Thus both strict sides of the level join their respective bands, and then the outer top and bottom edges, without meeting the level.

The compact set \{H_N=t\} blocks all top-to-bottom paths by continuity and lies in \cQ. By Lemma 6(ii) it contains a lateral continuum C_{N,t}; take d_1 to be the distance between the lateral edges of \cQ. Complete any straddling path to a top-to-bottom path using the strict-side connections. The completed path meets C_{N,t} by Lemma 6(i), and the added pieces avoid the level, proving the last assertion.□

Proof. Join two points of \overline B(x,Jr_N) inside B(x,LJr_N) by Lemma 4. Vertices of triangles meeting this path lie in B(x,(LJ+1)r_N); any two such vertices join inside B(x,L(LJ+1)r_N)\subset U for large N. The raw endpoint inequality bounds every difference of their values by the right side of (3.4), with C(J)=L(LJ+1)+3. It therefore bounds the interpolant’s oscillation along the original path. Apply (3.3).

Squaring and summing costs a bounded factor by Lemma 3: both the number of nearby weights and the number of centers charging each weight are bounded. The same holds for br_N-separated centers, with constants depending on b>0. A fixed g is bi-Lipschitz, so inverse images of target balls lie in source balls of radius C_gJr_N, their centers are b_gr_N-separated, and all these source balls lie in a fixed compact subset of U. The summation bound and (3.2) finish the proof.□

Proof. Choose R_* with 4R_*<d_1 and all closed 5R_*-balls about K contained in U. All following restrictions on m_0 concern only r_N/R=a^{-(N-l)}.

Expand at x,l by h, and choose macroscopic admissible weights \omega at scale N-l with squared mass M(N-l). Put

where C_0>C_E+2. Expansion on B(x,3R) separates the hq by a fixed multiple of r_{N-l}. Packing bounds both overlap multiplicities, giving

Let I be the open interval between the endpoint values. For t\in I, the continuum C_{N,t} meets the given path and exits B(x,2R), since its diameter exceeds 4R. By Lemma 4, join these two continuum points in its r_N/4-neighborhood and retain a radial subpath \eta_t between radii R and 2R, contained in that closed annulus. Its diameter is at least R, so h\eta_t is macroscopic by the choice of d_0 in Section 2.

If a target cell B(\xi,2r_{N-l}) meets h\eta_t at hy, choose q\in T_N with d(q,y)\leq r_N. For large m_0, this center belongs to \mathcal A and d(hq,hy)\leq C_Er_N/R, so \omega_\xi occurs in \sigma_q. Some level point lies within r_N/4 of y, hence in \overline B(q,2r_N). Thus t lies in the closed range interval I_q of H_N on that ball. Accounting for all target cells gives

The second inequality follows by integration and Cauchy–Schwarz. Apply (3.4) with J=2 and its bounded-overlap summation. Taking C(2)r_N<R/4 confines every charged center to R/4\leq d(q,x)\leq4R, proving (3.5).□

Proof. Fix compact K\subset U, and let x,y\in K with d(x,y)\leq r_k. They join inside B(x,Lr_k). Choose b with a^b>2L and consider indices l\leq k-b above a fixed interior threshold, in an arithmetic progression of step b_1 with a^{b_1}>16. The annuli \{r_l/4\leq d(x,\cdot)\leq4r_l\} are disjoint. For N sufficiently large relative to k, (3.5) and (3.1) give

Summing over at least ck indices yields |H_N(x)-H_N(y)|\leq C_Kk^{-1/2}. If \lambda>1, the single choice l=k-b gives C_K\lambda^{-k/2} instead.

For each tolerance this controls a tail uniformly; the finitely many omitted functions are individually uniformly continuous. Thus the sequence is equicontinuous on K. Arzelà–Ascoli on a compact exhaustion of U, followed by weak compactness of the measures on Z, gives the asserted subsequence. The two constant band values prove nonconstancy. The stronger estimate gives local Hölder exponent \log\lambda/(2\log a).

For (3.6), fix A>2L and choose As\leq R=r_l<aAs. Any two points of \overline B(x,s) join in B(x,Ls)\subset B(x,R/2). The annular and record estimates give

for large N. Local uniform convergence passes the oscillation to the limit; the closed-set inequality for weak convergence bounds the right-hand limsup by C\mu(\overline B(x,4aAs)). Take C_\mu=4aA.

For each fixed t, take a Hausdorff limit C_t of C_{N,t} in \cQ. It is connected and has diameter at least d_1. Local uniform convergence gives u=t on C_t\cap U. A fixed compact straddling path also straddles t for every sufficiently large H_N, hence meets those C_{N,t} and their limit. The subsequence used here may depend on t; no simultaneous selection is needed.□

Proof. Let I_q(f) be the closed range interval of f on the closed cell. For large n these cells lie in V. The interval w(D_t) is covered by the range intervals of w on cells meeting D_t, and each such cell has t\in I_q(f). Therefore

Integrate over I and use |I\cap I_q(f)|\leq\osc_{\overline B(q,2r_n)}f. Only finitely many interval indicators are integrated, so no measurable selection of D_t is required. Cauchy–Schwarz gives the final assertion.□

Proof. In a compact metric space X, a point’s component is the intersection Q_x of its clopen neighborhoods. To verify connectedness of Q_x, suppose it is separated into two compact pieces in disjoint open sets O_1,O_2. Compactness of X\setminus(O_1\cup O_2) gives a finite intersection W of clopen neighborhoods of x contained in their union. Its intersection with the O_i containing x is clopen in X and misses the other piece of Q_x, a contradiction. The same compactness argument shows that every open neighborhood of a component contains a clopen neighborhood of it.

Apply this to X=D\setminus B(p,r) and the component C of x. If C missed the sphere, it would have a clopen neighborhood W in X missing the sphere. Compactness places W a positive distance from \overline B(p,r), so W is also clopen in D. It is nonempty and misses D\cap B(p,r), contradicting connectedness. Thus C is the required subcontinuum.□

Proof. Write \delta(x)=\dist(x,Z\setminus O) and fix \eta>Q. A finite cover of \overline O gives a uniform Hölder bound, with some exponent \alpha_0>0, on a neighborhood of \overline O. For all large k put

The balls lie in \Omega, and

For the lower bound, cover a compact variation path P\subset O by at most C r_k^{-Q} radius-r_k balls centered on P, using Lemma 3. Their range intervals cover the interval f(P), so one ball has oscillation at least c r_k^Q; also \min_P\delta>0. The upper bound is the Hölder estimate.

Fix 0<\alpha<\alpha_0. Since A_k a^{\alpha k}\to0, there are arbitrarily late indices attaining their maximum over the remaining tail. At these indices,

Choose x_k\in O attaining at least half the supremum, and set \delta_k=\delta(x_k) and \sigma_k=A_k/\delta_k^\eta. Boundedness of f and (4.2) give \delta_k^\eta\ge c r_k^Q, hence \delta_k/r_k\ge c r_k^{Q/\eta-1}\to\infty. Expand at (x_k,k) and take a subsequence with p_k=h_kz_0\to b.

For a compact K\subset Z\setminus\{b\}, (2.3) gives

eventually. The expansion formula gives d(h_k^{-1}z,h_k^{-1}w)\le C_Kr_kd(z,w) on K. On every fixed multiple of this source ball, \delta is comparable to \delta_k. Thus (4.3) implies

For sufficiently small d(z,w), choosing j\ge0 with a^{-j}\ge C_Kd(z,w)>a^{-j-1} proves a uniform local \alpha-Hölder estimate for F_k on K.

For boundedness, Lemma 4 joins h_k^{-1}z to x_k inside a fixed multiple of B(x_k,r_k). Packing covers this path by a bounded number of radius-r_k balls centered on the path, each with oscillation at most C\sigma_k. Their range intervals cover the path’s image, so |F_k(z)|\le C_K. Compact exhaustion and Arzelà–Ascoli now give a locally uniform, locally Hölder limit. Finally, \osc_{h_k\overline B(x_k,r_k)}F_k\ge1/2. The lower puncture bound places all these image balls in one compact set off b, so local uniform convergence makes the limit nonconstant.□

Proof. For j\ge4 put K_j=\{d(z,b)\ge1/j\} and L_j=\{d(z,b)\ge1/(2j)\}. Choose the jth expansion so that h_j^{-1}L_j\subset U and its normalized u-transform differs from W by at most 1/j on L_j. Hold this expansion and its normalization fixed. Then choose N_j>N_{j-1} so large that

The uniform approximation follows from Proposition 2. With h_j and \sigma_j fixed, Lemma 9 bounds the displayed energy by C(h_j,K_j,j)/(\sigma_j^2M(N_j)), so the energy condition follows by increasing N_j. The last condition keeps all the enlarged balls in L_j. This proves local uniform convergence and vanishing energy at every fixed enlargement off b.

For a fixed g, choose a compact neighborhood K' of g^{-1}K avoiding b. Pulling back each target ball and choosing a nearest source net point gives, by bi-Lipschitz distortion and packing,

for some fixed J'\ge2 and all large n. Each source center receives only boundedly many target centers. Eventually K'\subset K_j and J'\le j, proving (4.6).

Uniform convergence makes the endpoint values of f_j straddle I. The source level \tau_j(t)=u(x_j)+\sigma_jt therefore lies in (0,1), and Lemma 8 supplies D_{j,t}=h_jC_{N_j,\tau_j(t)}, meeting P. Its source continuum has diameter at least d_1, so contains a point at distance at least d_1/2 from x_j. By (2.3),

uniformly for t\in I.□

Proof. Suppose x\in\mathcal N(W)\setminus\{c\} and V(x)\ne V(b). Choose \varepsilon>0 and a coordinate disk about x, compactly contained off b,c, on which |V-V(b)|\ge4\varepsilon. It contains a compact variation path P for W. Choose a compact positive-length interval I strictly between its endpoint values, avoiding W(c). Continuity gives \epsilon_c>0 such that \overline B(c,\epsilon_c) avoids b and its W-range has positive distance from I. Consequently, for all large j, the continua D_{j,t} from Lemma 14 avoid this closed ball.

Choose \rho>0 so that \overline B(b,\rho) avoids c,P and |V-V(b)|\le\varepsilon there. For w_j=f_j\circ g^{-1}, local uniform convergence gives

for all large j. Each D_{j,t} meets P and enters B(b,\rho). By Lemma 12, it has a subcontinuum joining a chosen point of P to the radius-\rho sphere and contained in

This subcontinuum lies in the level f_j=t and has w_j-oscillation at least \varepsilon. A fixed compact neighborhood of K avoiding both punctures eventually lies in h_jU\cap gh_jU and contains every closed radius-2r_{N_j} cell meeting K. Hence Lemma 11 and Cauchy–Schwarz give

a contradiction. All sets and g are fixed in this limit. Finally apply (4.7) at s^{-1}x with group element s^{-1}t to obtain (4.8).□

Proof. Let \xi be a based ray to b. For q\in E, |q|\le t, thinness of the ideal triangle with sides [e,q],\xi,q\xi puts q\xi(n) uniformly near \xi when n is sufficiently larger than t. The nearby parameter lies in [n-t-C,n+t+C]. For one common integer n, these vertices are distinct as q varies. Bounded valence therefore gives

If another accumulation direction exists, take q_n\to x\ne b and q_n^{-1}\to y along a subsequence. Base change gives, uniformly on compact sets avoiding y,

By Lemma 2, q_nz\to x there, and q_nb=b forces y=b. Choose disjoint small neighborhoods of b,x, with a point outside their union. For large n, q=q_n sends the complement of the first into the second. Its positive iterates never return the outside point, so q has infinite order. It is therefore loxodromic, with endpoints b,x'; see [9, III.\Gamma.3.10].

If some h\in E moved x', the cyclic subgroups generated by q and hqh^{-1} would intersect trivially: a nonzero common power would have the same pair of endpoints. Their m^2 products q^i(hqh^{-1})^j, 1\le i,j\le m, would then be distinct and have length O(m), contradicting (4.9). Thus every element of E fixes both endpoints. It maps a geodesic line between them to a line at uniformly bounded Hausdorff distance, so its orbit of e stays a fixed distance from that line. Its only possible accumulation directions are the two endpoints.□

Proof. Let E=\Stab(b) and let \Lambda_E be its at-most-two accumulation directions. Choose a variation region O for W whose closure avoids \{b\}\cup\Lambda_E. Such a region exists: otherwise W would be locally constant on the connected sphere minus this finite set, and continuity would make it constant throughout its domain.

Apply Lemma 13 to W on Z\setminus\{b\}, with this O and z_0=b. Write the expansions as s_i=v_i^{-1}, with centers y_i\in O, depths \ell_i\to\infty, and punctures p_i=s_i b\to b_*. Positive affine normalizations of W_i=W\circ s_i^{-1} converge locally uniformly off b_* to a continuous nonconstant W_*.

The punctures have a pairwise distinct subsequence. Indeed, if v_i^{-1}b=v^{-1}b infinitely often, then v_i v^{-1}\in E. After a subsequence y_i\to y\in\overline O. Since v_i lies at depth \ell_i on a ray to y_i, Lemma 2 gives v_i\to y in boundary direction. Bounded right multiplication preserves this direction, implying y\in\Lambda_E, a contradiction.

Choose two coordinate disks D_1,D_2, with compact closures off b_*, having disjoint W_*-range intervals and each containing a compact variation path. To do this, follow a variation path to its first hitting points of two distinct intermediate values. Just before either first hitting time the values differ, so the function is not locally constant at either point. Continuity supplies disks with disjoint ranges, and non-local constancy supplies a variation path in each.

For every sufficiently large i, the puncture p_i is outside the closed disks. Uniform convergence of the normalized W_i preserves both the disjoint range intervals and the positive variation on the two fixed paths. Positive affine normalization preserves these properties, so \mathcal N(W_i) meets each disk. Fix two distinct such indices i,j. By (4.8), W_j has the single value W_j(p_i) at every point of \mathcal N(W_i) in either disk. Taking one point in each contradicts the disjoint W_j-range intervals.□

Proof. Fix b\in[AR,B_0R]. Each C_t, t\in J, meets the radius-b sphere, so the inner intervals of candidates with |d(x,y)-b|\leq r cover J. By (5.3), two of them meet the respective quarters in (5.5) by at least \kappa\Delta.

If both have t_y<1/8, the lower-quarter outer interval ends below j_-+3\Delta/8, and the upper-quarter interval starts above j_-+5\Delta/8. Indeed each meets its quarter but overlaps J in length less than \Delta/8; the assertion also holds for intervals extending beyond J. Their distance exceeds \Delta/4, making the node good. Otherwise one candidate has t_y\geq1/8. Apply this at radial values spaced by (H+3)r: there are at least c_Ha^D such values, and the resulting centers are more than Hr apart.□

Proof. Join two candidates in a graph when their distance is at most Hr. Packing bounds its degree, so graph coloring partitions Y into at most C_H separated classes, with C_H independent of D. If the conclusion fails, then

Call y large when t_y\geq a^{-(\tau+\gamma)D} and small otherwise. On large candidates the minimum is linear, so

We will test levels at an intermediate radius s, larger than the child radius r and smaller than the parent radius R. This provides more radial tests than the large ranges can occupy. At the same time, it lets us detect the remaining levels with a modulus cost at the relative scale r/s.

Remove the large ranges. Set D'=\lceil\beta D\rceil and s=a^{-D'}R. For large D, 0<D'<D and s<R/4. Choose s-spaced radial bins \mathcal B in [(A+1)R,(B_0-1)R], so \#\mathcal B\asymp a^{D'}. On \mathcal B\times J use counting measure times Lebesgue measure. Mark (b,t) if a large candidate satisfies

Each candidate meets at most 11 bins. By (5.4) and (5.13), the marked measure is at most CC_Ha^{\gamma D}\Delta. Since \gamma<\beta, the unmarked measure is at least

Detect the remaining levels by modulus. Take a maximal s-separated set \mathcal Z_s in \mathcal A_x(R); its radius-s balls cover that annulus, and it has at most Ca^{QD'} points. Expand at each z\in\mathcal Z_s at depth k+D' by h_z. Put q=r/s=a^{-(D-D')}, and take macroscopic admissible weights \omega_\xi, \xi\in T_{D-D'}, of squared mass M(D-D'). For fixed C_T>2+C_E, define

Within \overline B(z,3s) the candidate images are separated by a fixed multiple of q. Thus every sum has boundedly many terms and every target weight is charged boundedly many times. Thus \sum_y(\sigma_y^z)^2\leq CM(D-D') for each z. Also each y lies within 3s of boundedly many auxiliary centers. Packing and Cauchy–Schwarz give

Fix an unmarked (b,t). Choose w\in C_t on the radius-b sphere and z\in\mathcal Z_s with d(w,z)\leq s. The diameter bound gives a point of C_t outside \overline B(w,2s). Join these points in the r/4-neighborhood of C_t, by Lemma 4, and stop at the first radius-s hit about w. The resulting path \pi has diameter at least s and lies in \overline B(w,s)\subset\overline B(z,2s). The scale-s expansion h_z therefore makes it macroscopic by the choice of d_0 in Section 2.

The radial bin lies between (A+1)R and (B_0-1)R, while s<R/4. Thus every point of \pi lies between radii (A+3/4)R and (B_0-3/4)R about x. In particular it lies in the candidate annulus and has a candidate center within distance r. The same fixed source margin places every level point within r/4 of \pi in K^+\subset U. On these points the scalar-limit property gives u=t, even though it does not assert this identity on parts of C_t outside U.

For a target cell meeting h_z\pi, choose a preimage hit v\in\pi and y\in Y with d(y,v)\leq r. Then d(y,z)<3s. If \xi is the target cell’s center, (2.2) gives d(\xi,h_zy)<(2+C_E)q<C_Tq, so its weight occurs in \sigma_y^z. A point of C_t is within r/4 of v, hence within 5r/4<c_0r of y. The preceding interior check gives t\in I_y^{\mathrm{out}}. Since |d(x,y)-b|\leq s+r<5s, unmarkedness makes y small. Admissibility therefore yields

The unmarked levels now have an admissible detection weight supported on small candidates. We compare its squared mass with the surviving amount of level parameter.

Compare the energies. Integrating (5.16), using bounded bin multiplicity, gives

Only finitely many closed-interval indicators are integrated; no measurable selection of continua or paths is needed, and singleton ranges contribute zero. Cancel \Delta>0. Cauchy–Schwarz and (5.15) imply

Subexponential growth gives M(D-D')\leq a^{\tau D/4} for large D: first bound M(j)\leq Ca^{\tau j/8} and then absorb C by increasing D. Since D'\leq\beta D+1 and (Q-2)\beta<\tau/4,

On small candidates this is exactly the capacity, contradicting (5.12). Every threshold for D is uniform over nodes.□

Proof. By (5.21), (5.20), and Jensen’s inequality,

Thus

Conservation of probability identifies the left side with the average bad mass over the first n generations. If good mass were eventually less than 1/2, that average would have lower limit at least 1/2, a contradiction.□

Proof. Centers assigned to the same depth-k vertex have mutual distance O(R). Their HR-separation and Lemma 3 bound their number. If several puncture balls share a point, their punctures have mutual products at least k-O_C(1). By (2.4) and Lemma 2, all corresponding inverse vertices lie in one bounded Cayley-graph ball. There are boundedly many such vertices, each with the bounded multiplicity just proved.□

Proof. Fix an ancestor A at generation n-m, whose depth is k-l=k_0+D(n-m). We claim that only boundedly many descendants j have d(p_j,p_i)\le a^{-l}. Split a geodesic word

The puncture condition and (2.4) give (p_j|p_i)_e\ge l-C and (v_j^{-1}|p_j)_e\ge k-C. Thus w_j lies a bounded distance from the depth-l point on a fixed ray to p_i, leaving boundedly many possibilities.

Reversing and inverting the same word gives the geodesic v_j=(u'_j)^{-1}w_j^{-1}. Its depth-(k-l) prefix is therefore (u'_j)^{-1}. Since

thinness and Lemma 2 put this prefix a bounded distance from the depth-(k-l) point on a fixed ray to x_A. Hence u'_j also has boundedly many possibilities. Their products determine boundedly many vertices v_j, and the first step of Lemma 20 bounds the nodes assigned to each. We count group elements here, so nonuniqueness of geodesic spellings causes no additional multiplicity. The source and inverse prefixes together account for the full word.

Each such descendant has P_j\le P_Aa^{-\gamma Dm} by (6.1). Sum first over these descendants and then over ancestors, whose probabilities sum to one.□

Proof. Write F_i=u\circ h_i^{-1} and F_j=u\circ h_j^{-1} on their open domains. If \osc_{E_i}F_j\ge\Delta_j/16, endpoint j qualifies. Here we use \kappa\le1. Otherwise, one pattern ball \mathcal B^s has its F_j range interval at distance at least \Delta_j/16 from that on E_i. Indeed, if neither did, the two pattern intervals would have distance less than 3\Delta_j/16, contradicting their separation by \Delta_j/4.

Node i supplies a compact path \eta\subset\mathcal B^s with endpoint F_i-gap at least \kappa\Delta_i. Suppose \osc_{E_j}F_i<\kappa\Delta_i/4. Removing this latter range interval from the endpoint gap leaves a positive-length closed interval I\subset(0,1) strictly between the endpoint values and a positive distance from the range on E_j. To obtain positive margins, choose a smaller closed interval inside one of the remaining open intervals. Its length need not be uniform over pairs.

Fix the pair and I before letting N\to\infty. Put F_{i,N}=H_N\circ h_i^{-1} and F_{j,N}=H_N\circ h_j^{-1}. The fixed range margins and local uniform convergence imply, for every t\in I and all sufficiently large N, that h_iC_{N,t} hits \eta and avoids E_j. It also reaches the inner i-puncture ball. For the first assertion, the endpoint values still strictly straddle all of I, so Lemma 8 applies to the preimage path. For the second, the range on E_j remains disjoint from I. By Lemma 12, it has a subcontinuum K_{N,t} containing a hit on \eta, meeting the sphere d(z,p_i)=A_hR, and avoiding B(p_i,A_hR). It remains in the level F_{i,N}=t and avoids E_j. In particular,

Compact containment follows from (6.6) with half the radius. The sphere hit belongs to E_i; no identification of that sphere with the boundary of a metric ball is needed.

Uniform convergence on \eta\cup E_i preserves the F_j range gap, giving \osc_{K_{N,t}}F_{j,N}\ge\Delta_j/32 uniformly for t\in I. The finite-cell comparison, Lemma 11, now gives

For large N these closed cells lie in the common domain. Both squared oscillation sums tend to zero by Lemma 9, so Cauchy–Schwarz contradicts the positive left side. Therefore \osc_{E_j}F_i\ge\kappa\Delta_i/4, proving the alternative. The argument applies separately to every pair above the geometric threshold; no convergence rate uniform over pairs is required.□

Conclusion of the proof of Proposition 5. Assign each unordered far pair to a qualifying endpoint. At any point, (6.12) allows only Ca^{Ql} paying indices i, and (6.11) allows boundedly many partners for each. Integrating this multiplicity bound gives

This counts the enlarged balls directly and uses no invariance of \mu. The assignment may be arbitrary when both endpoints qualify: all lower bounds hold for either endpoint, and the multiplicity estimate already counts every possible ordered contribution.

Conversely, every paying endpoint satisfies

There are at least c\mathcal N^2 far pairs, so (6.10) and (6.3) give total cost

Since \kappa,\Delta_0>0 are fixed and Ql=QD\lfloor\gamma n/Q\rfloor\le\gamma Dn, comparison with (6.13) yields

This contradicts k=k_0+Dn, excluding the unbounded subexponential case.□

Proof of Theorem 1.1. If M(n) were unbounded, the record selection of Section 3 would give either exponential records, excluded by Proposition 3, or subexponential growth, excluded by Proposition 5. Thus \sup_n M(n)<\infty. Theorem 2.1 uniformizes the boundary, and Proposition 1 supplies the proper cocompact action with finite kernel.□

Proof. Point stabilizers in a proper action are finite, hence trivial when G is torsion-free. Thus the action in Theorem 1.1 is free, and \HH^3/G is a closed hyperbolic three-manifold with fundamental group G. The manifold may be nonorientable.□

Proof. Let M^+ be the orientation cover of the closed hyperbolic manifold in Corollary 1, using the manifold itself when it is orientable, and let G^+=\pi_1(M^+). Thus G^+ is normal of index at most two in G. Agol’s virtual-fibering theorem [1, Theorem 9.2] gives a connected finite cover N of M^+ fibering over the circle with closed connected orientable fiber \Sigma. The fibration sequence gives \pi_1(N)\cong\pi_1(\Sigma)\rtimes\ZZ, with action induced by the monodromy, and its surjection onto \ZZ gives b_1(\pi_1(N))\ge1.

Bergeron–Wise’s cubulation theorem [5, Theorem 5.3] gives a proper cocompact action of G^+ on a CAT(0) cube complex, free because G^+ is torsion-free. Agol’s virtual-special theorem [1, Theorem 1.1] then gives a finite-index subgroup of G^+ with compact special quotient, so G is virtually compact special. Agol’s separability corollary [1, Corollary 1.3], applied to the trivial subgroup, also makes G^+ residually finite. The finite quotient G/G^+ and intersections of the finitely many G-conjugates of finite-index normal subgroups of G^+ then separate every nontrivial element of G.□