The ℓ¹-Bass Conjecture for Discrete Groups

1 Introduction

Let G be a discrete group. Its complex Banach group algebra \ell^1(G) consists of functions a\colon G\to\C with \|a\|_1=\sum_{g\in G}|a(g)|<\infty, equipped with convolution

Write \mathcal C(G) for the set of conjugacy classes, and \mathcal C_{\rm fin}(G) for those consisting of finite-order elements. The subscript refers to element order, not to the size of a conjugacy class.

For an idempotent matrix e=(e_{ij})\in M_q(\ell^1(G)), its Hattori–Stallings trace [7, 13] has coefficients

These sums converge absolutely, and \sum_C|\tau_C(e)|\le\sum_i\|e_{ii}\|_1. The trace is additive under direct sums and depends only on the isomorphism class of the projective module e\ell^1(G)^q. It therefore defines

where K_0 is the Grothendieck group of finitely generated projective right modules. The \ell^1-Bass conjecture [2, Conjecture 2.2] asserts that this map takes values in the algebraic direct sum over \mathcal C_{\rm fin}(G). We prove this assertion for all discrete groups.

The finite-support assertion is part of the conclusion, in addition to vanishing on infinite-order classes. It is not an automatic consequence of absolute summability. In particular, the proof must make its geometric cutoff independent of the conjugacy class under consideration.

1.1 History and relation to earlier methods

The Hattori–Stallings trace refines the rank of a projective module by taking values in the quotient of its ring by additive commutators [7, 13]. For the algebraic group rings \Z G and \C G, this quotient is indexed by conjugacy classes. Bass’s conjecture for the integral group ring asks that this trace be supported at the identity [1]; the complex-group-ring formulation permits finite-order classes. These support questions connect the algebraic structure of projective modules with the conjugacy structure of a group.

Cyclic homology gives one route to these questions. Burghelea’s computation separates group-ring cyclic homology into contributions from finite- and infinite-order conjugacy classes [3]. Eckmann used this description and the Chern character to prove Bass-type vanishing for several classes of groups [5]. Emmanouil extended this method through nilpotence properties of the cyclic periodicity operator [6]. Our argument retains cyclic symmetry but works directly with weighted configurations and differential forms.

Berrick, Chatterji, and Mislin formulated the \ell^1-Bass conjecture in the finite-support form used here. They proved it for countable groups whose Bost assembly map is rationally surjective in degree zero [2, Theorem 1.4]. Together with Lafforgue’s Banach-algebra K-theory [11], this includes amenable groups; the passage from countable subgroups to arbitrary groups is explained in [2, Lemma 1.5]. Theorem 1.1 removes the assembly hypothesis while retaining the algebraic finite-support conclusion.

Ji, Ogle, and Ramsey developed related Bass conjectures for rapid-decay completions and higher Chern characters [9, 10]. Their work highlights the growth estimates needed when passing from group rings to analytic completions. Here the summability needed in the proof comes from a similarity reduction to an exponentially decaying idempotent. The subsequent estimates use a spatial scale uniform over all classes to be excluded.

The weighted cyclic configurations and Pfaffian detection strategy build on the complex-group-ring argument in [12, Sections 7–8]. In the group-ring setting, nonzero products of coefficients involve group elements from a fixed finite set. For an \ell^1 idempotent these elements may have arbitrarily large word length, even after reduction to exponential decay. The estimates must retain summable length factors without losing control as the number of factors increases. We obtain them from smooth spatial summaries whose derivative bounds are independent of the size of the input support, multiscale time averaging with a positive translation component, and a final ordered-kernel bound uniform in the closing group element. A single spatial radius then excludes all but finitely many finite-order classes. All ingredients used here are proved below; the complex-group-ring theorem is not an input. Standard concentration and transgression arguments are credited at their points of use.

1.2 Proof strategy

The argument uses Banach algebra power series, differential forms on finite-dimensional simplices, and elementary estimates for integral operators. We give the needed estimates in full.

First, a finite-support approximation followed by an analytic idempotent correction replaces e by a similar idempotent p with an exponential moment in a finitely generated subgroup (Section 2). This step preserves every trace coefficient. It does not assert that p has finite support.

For a fixed g\in G, consider cyclic lists of group elements whose closing edge is twisted by multiplication by g. Products of the coefficients of p assign complex weights to these lists. Their total weight is \tau_{[g]}(p), and idempotency makes the weights compatible with deleting vertices. The ordered times of a list form a simplex. On these simplices, a one-form invariant under simultaneous time translation and equal to one on the translation vector will be called a connection form. A cyclic Stokes argument shows that the weighted average of its curvature Pfaffian equals the trace coefficient, up to sign (Section 3).

The task is then to construct compatible connection forms with small averaged curvature. Interpret a timed list as a step function with values in G, and average its Dirac masses on a sequence of time scales. The resulting probability distributions are summarized by smooth subprobability distributions on auxiliary labels. These summaries record spatial concentration while controlling derivatives caused by moving one boundary of the step function (Section 4). Pairing their differentials with bounded odd time kernels gives one-forms. Their contractions with time translation telescope across the averaging scales. A final equality kernel detects the twist and leaves a strictly positive contraction, which allows normalization to a connection (Section 5).

The curvature estimates use three different features of this construction. At the initial scale, a curvature entry vanishes unless its two times are close; an occupancy estimate controls the resulting sparse Pfaffians on average. At intermediate scales, the short support of the time kernel gives a Hilbert–Schmidt bound. At the last scale, the equality kernel reduces to weighted order kernels, whose singular values satisfy s_\nu\le C/\nu. Expanding the Pfaffian across these stages gives exponential decay in the simplex dimension (Section 6). A single spatial radius works for every infinite-order class and every finite-order class avoiding a fixed finite word ball. This uniformity proves the finite-support conclusion in Section 7.

2 Reduction to exponential decay

The first step replaces a general \ell^1 idempotent by an idempotent whose coefficients have exponential decay in a finitely generated subgroup. The replacement preserves every trace coefficient exactly. This permits the polynomially weighted sums that arise from differential forms in arbitrarily large, but fixed, dimensions.

For a complex matrix A=(A_{ij}), write |A|_1=\sum_{i,j}|A_{ij}|. On M_q(\ell^1(G)) we use

This norm is submultiplicative. The identity matrix has norm q; none of the power-series arguments below requires its norm to be one.

The same rearrangement applies to rectangular matrices: if a\in M_{q,r}(\ell^1(G)) and b\in M_{r,q}(\ell^1(G)), then T_C(ab)=T_C(ba), with the trace in the appropriate size on each side. An isomorphism between two finitely generated projective modules, together with its inverse, can be extended by their defining idempotents to such matrices a,b with products equal to the two idempotents. This proves the module invariance used in the definition of \operatorname{HS}^1.

Henceforth G is finitely generated, d(x,y)=|x^{-1}y| is a fixed left-invariant word metric, and p satisfies (2.1). We fix a constant \kappa_G such that

Such a bound follows by counting words in the generators. It imposes no additional growth hypothesis on G. Constants without a subscript may depend on this metric and on fixed construction choices. A constant denoted C_R may also depend on the radius parameter R introduced later, but will not depend on a tuple, its twist g, or the dimension n.

3 Cyclic simplex formulas for the trace

We now express a conjugacy-class trace coefficient as a weighted average of curvature Pfaffians. Idempotency provides compatibility of the weights under deletion of a vertex; cyclicity supplies the cancellation that makes the average independent of the connection form. The tuple weights have the same algebraic origin as the weighted paths in [12, Section 7]; we give the identities needed here directly.

3.1 Tuples and their weights

Fix g\in G and write H=C_G(g) for its centralizer. For k\ge0 let

where H acts by simultaneous left multiplication. For a tuple h=(h_0,\ldots,h_k) extend its indices to all integers by h_{i+k+1}=gh_i and define

The product is in the displayed order. Both expressions descend to \mathcal T_k(g), and l_i is periodic with period k+1.

3.2 Compatible forms

For each tuple use the ordered time domain

Extend times by t_{i+k+1}=t_i+1, and set x_i=t_{i+1}-t_i for 0\le i\le k. Thus x_i\ge0 and \sum_i x_i=1. Simultaneous time translation is generated by V=\sum_{i=0}^k\partial_{t_i}. Its quotient is the simplex

using the section t_0=0, oriented by dt_1\wedge\cdots\wedge dt_k. We regard [t_i,t_{i+1}) as carrying the label h_i. The face x_i=0 corresponds to deleting (h_i,t_i): the interval carrying the label h_i has length zero. At the closing face use the periodic lifts to make this same convention.

Translation invariance and \alpha(V)=1 imply \iota_Vd\alpha=0. Thus d\alpha is a form on \Delta_k independent of the choice of translation section. A translation-invariant form whose contraction with V is zero will be called basic.

For a real skew-symmetric 2m\times2m matrix A, our convention is

In particular \Pf(A)^2=\det A. This identity follows by reducing a skew form to two-dimensional blocks, with both sides transforming by the square of the determinant of a change of basis.

The proposition reduces trace vanishing to a quantitative construction. We shall choose connections, in arbitrarily large even dimensions, for which the average of the absolute Pfaffian is at most a small exponential factor times \prod_{i=1}^n l_i. Equation (3.2) then permits summation against the tuple weights. The next section constructs the spatial summaries used to obtain these connections.

4 Smooth summaries at a spatial scale

The connection forms will be built from smooth summaries of probability distributions on the group. A summary must retain information about spatial concentration while changing in a controlled way when mass moves between two group elements. We construct two versions. The first allows the spatial radius to increase by a factor of six and has a derivative bound independent of that radius. The second retains the radius and has a polynomial dependence on it. Their different bounds will be used at different stages of the averaging construction.

Throughout this section, G is a finitely generated group with a fixed left invariant word metric d. Write B(y,s)=\{h:d(h,y)\le s\}. Counting words in a finite symmetric generating set gives a constant c_d such that

Indeed, if the generating set has m elements, the ball is covered by words of length at most \lfloor s\rfloor, and their number is at most (m+1)^{\lfloor s\rfloor+1}. Increasing c_d also covers the trivial group. Constants denoted by C may depend on this word metric but not on a radius, an accuracy parameter, or the size of an input support.

Let \mathcal P_f(G) be the set of finitely supported probability distributions on G. For r\ge1 set

The maximum exists: the score vanishes outside the finite 2r neighborhood of \supp P. It lies in [0,1], so 0\le\phi_r\le0.20. The functions q_r and \phi_r need not be smooth; they will only be used in inequalities.

In a transfer direction \dot P=\delta_h-\delta_{h'}, derivatives are computed after enlarging S to contain h,h'. The channels we construct will have compatible formulas on these enlarged spaces, and their derivatives will vanish at zero output coordinates. For such a channel define its Fisher seminorm in this direction by

The zero-coordinate assertion is part of the construction, and will be proved; it is not a convention for discarding a nonzero derivative.

If A has label set G and L has label set \Lambda, a cross kernel is a function \kappa:G\times\Lambda\to[0,1] satisfying \kappa(sy,s\omega)=\kappa(y,\omega). It is fixed independently of P. It gives the predictions

Exact predictions alone would be easy: L(P)=P and \kappa(y,h)=\chi_r(d(h,y)) give \operatorname{pred}(P,y)=E_r(P,y). When G has at least two elements, this choice has no uniform Fisher bound: a transfer between distinct elements, adding mass to an entry of size t>0, contributes 1/t to the squared seminorm, which diverges as t\downarrow0. The approximation allowed in the next proposition permits uniform derivative control, including at probability faces.

To see why these guarantees are suited to averaging, assume them for the moment. Take a finite family P^{(\nu)} of input probabilities and weights \theta_\nu\ge0 with \sum_\nu\theta_\nu=1, and write

The selection and lower prediction bounds give \sum_yA(P)_y(\operatorname{pred}(P,y)-a_*)\ge\phi_r(P)-20\epsilon. On the other hand, the upper prediction bound and linearity of E_{r'} give \overline{\operatorname{pred}}_y \le E_{r'}(\overline P,y)+10\epsilon. Since \overline A is a subprobability, its pairing with E_{r'}(\overline P,\cdot)-a_* is at most \phi_{r'}(\overline P); if the maximal centered score is negative, that pairing is nonpositive. Finally, \sum_\nu\theta_\nu\sum_yA(P^{(\nu)})_y=\sum_y\overline A_y, so the terms containing a_* cancel and

Thus loss of concentration under averaging gives a lower bound for the difference of the two pairings. The same calculation applies to time averages of fields with a common finite support, which is how these estimates will enter the connection construction.

We first construct A and a second distribution D on possible centers. The channel A concentrates near a maximizing center when the maximal score exceeds a_*. The thresholds for D are lower, so that D already has almost full mass whenever A is nonzero. The two versions of L will then use this same D to predict scores at the selected centers.

4.1 Smooth selection of centers

The number of centers that can carry substantial score has a bound independent of |\supp P|.

Choose smooth functions \psi_A,\psi_D:\R\to[0,1] such that

For completeness, a smooth increasing cutoff from zero to one on [0,1] is \eta(t)/(\eta(t)+\eta(1-t)), where \eta(t)=e^{-1/t} for t>0 and \eta(t)=0 for t\le0. Every right derivative of \eta at zero vanishes: away from zero it is e^{-1/t} times a polynomial in 1/t, which tends to zero as t\downarrow0. The denominator in the displayed quotient is everywhere positive, and affine changes of variable give (4.10). Their first derivatives are bounded by a fixed numerical constant, since they vanish outside fixed compact transition intervals.

Take N=M_0r\epsilon^{-2} and define

Here M_0 is fixed in terms of the metric, large enough that for all allowed r,\epsilon,

For example, any M_0\ge25\gamma+4 suffices. Indeed, \log(1+e^{\gamma r})+\log(1/\epsilon) \le\gamma r+\log2+1/\epsilon\le(\gamma+2)r/\epsilon, and N\ge25\gamma r.

The additional 1 in each denominator may be regarded as the weight of an unused null label. It allows the channel to have small total mass when no center reaches the exponential threshold. Each displayed sum is finite by Lemma 4.3.

The final averaging stage will only need D, together with its concentration on a bounded set of labels. We record that consequence separately.

4.2 Predictions with an enlarged radius

The early channel can use the average score of the centers chosen by D. All these centers lie near any center selected by A. Enlarging the radius allows us to bound each score contributing at y by E_{6r}(P,y). Define

This is a subprobability channel, since 0\le E_r(P,z)\le1. When d(y,z)\le4r, every h with \chi_r(d(h,z))>0 has d(h,y)<6r. On this set \chi_{6r}(d(h,y))=1, and therefore

After summing against D, this proves \operatorname{pred}(P,y)\le E_{6r}(P,y).

If A(P)_y>0, then E_r(P,y)>0.70 and q_r(P)>0.70. For every z with D(P)_z>0, Lemma 4.3 gives d(y,z)\le4r. Hence every such z contributes to the prediction, and (4.14) gives

To estimate derivatives, write E_z=E_r(P,z) again. Where D_z>0, E_z>0.58. Differentiating L_z=D_zE_z yields

In the second sum only D_z>0 is relevant; its denominators are thus bounded below by 0.58. We used (4.16), (4.15), and r\ge1. At a zero coordinate D_z=0 and \dot D_z=0, so \dot L_z=0. The same finite formulas prove smoothness at faces. This establishes all the early-version claims of Proposition 4.2.

4.3 Accurate finite sample strings

To keep the radius equal to r, an average of center scores is insufficient: we must predict the score separately at each y. A finite string of sample points approximates all scores in a bounded region simultaneously. Random sampling will prove the existence of an accurate string. We will then put smooth positive weights on all strings in a fixed finite alphabet. This distinction is useful at probability faces: the defining weights will not be products of possibly vanishing input probabilities.

Fix a center z\in G. Introduce a null symbol \partial, fixed by the group action, and the alphabet

Define a probability Q^z on this alphabet by

The notation E_r(P^z,y) has the same linear meaning as in (4.2), although P^z can have mass less than one. This truncation provides a fixed finite alphabet while preserving every score needed for the lower prediction bound. Indeed, if A(P)_yD(P)_z>0, Lemma 4.3 gives d(y,z)\le4r. Every h contributing to E_r(P,y) then has d(h,z)<6r, where the factor \chi_{10r}(d(h,z)) is one. Consequently

For arbitrary y, we still have E_r(P^z,y)\le E_r(P,y), which is the direction needed for the upper bound. Thus accurate predictions for P^z will suffice. The Lipschitz taper in (4.23) will also keep their transfer derivatives proportional to d(h,h')/r.

For a positive integer b and a string w=(w_1,\ldots,w_b)\in\mathcal A_z^b let

Both scores vanish when d(y,z)>22r, by the triangle inequality. Thus \Delta also controls the discrepancy at every y\in G. Always 0\le\Delta\le1.

4.4 Smooth weights on strings

Fix the integer b supplied by Lemma 4.6. To assign smooth weights, replace the maximum of the signed errors by the logarithm of a sum of exponentials. Put

Choose \lambda>0, depending only on r,\epsilon, so large that the logarithm of the number of signed tests is at most \lambda\epsilon. The number of tests is independent of z, since left translation identifies the balls. The elementary inequalities e^{\lambda\max x_i}\le\sum_i e^{\lambda x_i} \le M e^{\lambda\max x_i} give

The first derivative of this smoothed maximum has a particularly simple bound. The test function h\mapsto\chi_{10r}(d(h,z))\chi_r(d(h,y)) is 11/(10r)-Lipschitz: both factors have absolute value at most one, and their Lipschitz constants add. Thus every signed error in (4.26) has transfer derivative of absolute value at most 11d(h,h')/(10r). Differentiation of the logarithm gives a convex combination of these derivatives; the factor \lambda cancels. Consequently

In particular this estimate does not grow with the number of tests or the smoothing parameter.

Now define a probability distribution on the entire fixed set of strings by

Every coordinate is strictly positive. The constant M_2 will be fixed in terms of the metric.

4.5 Predictions at the original radius

We can now finish the later channel. Its label set consists of pairs (z,w), with z\in G and w\in\mathcal A_z^b. Translation acts on the center and on every nonnull entry of the string. Define

The total mass of L is \sum_zD(P)_z\le1, and the kernel takes values in [0,1]. All formulas commute with left translation.

By (4.31), and since P^z_h\le P_h,

Suppose next that A(P)_y>0. For every z with D(P)_z>0, (4.24) identifies the predicted truncated score with E_r(P,y). Moreover q_r(P)>0.70, and (4.14) gives \sum_zD(P)_z\ge1-\epsilon. It follows that

These are the required prediction bounds with r'=r.

For the derivative bound it is useful to retain the product structure L_{z,w}=D_z\pi_z(w). Since \sum_w\pi_z(w)=1 and \sum_w\dot\pi_z(w)=0, expansion of the square gives the exact identity

All weights and derivatives here are real; the mixed term vanishes by the displayed zero-sum identity. Using (4.15), (4.32), and b\le Cr\epsilon^{-4} yields

It remains to check the regularity promised in the channel definition. For a fixed finite input support S, only finitely many centers z can have D_z\ne0, all in the 2r neighborhood of S. For each such center, the alphabet and string set are finite and independent of P. The functions S(P;z,w) and \pi_z(P)_w are smooth functions of the coordinates of P: the scores are linear, and each exponential sum has a strictly positive denominator. Thus (4.33) is a fixed finite smooth formula on each enlarged support space. If L_{z,w}=0, strict positivity of \pi_z(w) implies D_z=0; also \dot D_z=0 by Lemma 4.4. Consequently \dot L_{z,w}=0. This proves smoothness at faces and the required behavior at zero entries.

For use with differential forms, regard the labels of A and L as disjoint even when their underlying sets happen to coincide. On their disjoint union define the symmetric kernel K by

For vectors on this union, write K(U,V)=\sum_{\xi,\eta}K(\xi,\eta)U_\xi V_\eta. The combined field Y=A\oplus L has mass at most two, and

The kernel is bounded and invariant under simultaneous left translation. The expression in (4.3) also makes sense for this field of mass at most two, and its square is \mathcal F_A(P;h,h')^2+\mathcal F_L(P;h,h')^2. Thus Proposition 4.2 controls its derivatives as well. These facts permit both channels to enter a single bilinear pairing in the averaging construction.

5 A connection from successive time averages

Proposition 3.3 allows us to compute the trace with any connection satisfying Definition 3.2. We now construct a connection whose curvature can be estimated at several time scales. The construction first gives a one-form B with uniformly positive contraction B(V); division by this contraction will produce the connection. The summary channels of Proposition 4.2 make the contributions of successive scales telescope. The final contribution uses the twist by g.

Throughout this section G is finitely generated, d is its fixed left invariant word metric, and g\in G. Constants denoted by C may depend on the word metric and on the fixed smooth functions used below. After the parameter choice at the end of the construction, a subscript R will permit dependence on the radius R, but never on the tuple, its dimension, or the element g.

5.1 Time averages and spatial summaries

Begin with a finite number b_*\geq2 of averaging stages, an even ambient dimension n, and positive time scales \delta_1<\cdots<\delta_{b_*} satisfying

The construction through positivity uses only these conditions on the time scales. Their precise dependence on R and n will be selected afterwards, when preparing the quantitative estimates. The same scales are kept fixed in every dimension 0\leq k\leq n. Thus passing to a face never changes the averaging operation.

Choose a nonnegative, even C^\infty function \mu, supported in [-1,1], with integral one, and put \mu_j(s)=\delta_j^{-1}\mu(s/\delta_j). On a tuple of dimension k, write its step function and its averages as

Zero-length intervals are omitted. These are finitely supported probability distributions at each time, and P_j(u+1)=gP_j(u). If \eta_j=\mu_j*\cdots*\mu_1, then

The support bound follows from (5.1), and the supremum bound follows by convolving \mu_j with probability densities.

To summarize these averages spatially, choose R\geq1 and an integer 2\leq J_*\leq b_*. Put

The radius increases by a factor of six during the first J_*-1 steps and is then fixed at r. Use accuracies

where 0<\epsilon_*\leq10^{-3} is fixed sufficiently small. The summable errors will leave a positive quantity after telescoping.

For 1\leq j<b_*, apply Proposition 4.2 to P_j(u) at parameters (r_j,\epsilon_j). Use its early version when j<J_* and its later version when j\geq J_*. Denote the resulting channels by A_j,L_j and their prediction kernel by \kappa_j. On the disjoint union of the two label sets put

Define a symmetric bilinear kernel K_j to be zero between labels in the same channel and to have value \kappa_j(y,\omega)/2 between the two channels. Thus

The total mass of Y_j(u) is at most two, and |K_j|\leq1.

At the first and last stages use

where D is the channel of Lemma 4.5 at parameters (r,\epsilon_{b_*}). These two fields have mass at most one. All the fields obey Y_j(u+1)=gY_j(u), and every kernel is invariant under simultaneous left translation of its labels.

The last channel needs a multiplier to compensate for its possible spread over many labels. Lemma 4.5 gives \sum_zD(P)_z^2\geq\frac14e^{-C_Dr} when \phi_r(P)>0, for a fixed constant C_D. Choose a fixed C_*\geq C_D and set

Since 0\leq\phi_r\leq1-a_*=1/5, this gives, for every accuracy parameter in (5.5),

We shall construct the connection under either of the assumptions

Only the final time-lag kernel and its contraction distinguish these two cases.

5.2 Lag pairings and their compatibility

We next turn the fields into one-forms. For 0\leq j<b_* let \rho_j=\mu_{j+1}*\mu_{j+1} and define the odd function

It is bounded in absolute value by one and vanishes outside [-2\delta_{j+1},2\delta_{j+1}]. In the sense of distributions,

Here \delta_0 denotes the Dirac measure at zero, rather than a time scale. At the final stage set

Formula (5.13) holds at this stage as well. Values at the jump points have no effect on any integral below.

The differential dY_j(u) in the following definition acts on the tuple times t_0,\ldots,t_k, with u fixed. Write \int_{\rm per} for integration over one period, using a half-open interval when integrating the raw impulse measure, and put

For j=0 the differential of the step field is a measure in u. Specifically, set

The periodic convention gives h_{-1}=g^{-1}h_k. Thus the coefficient of dt_i in the raw form is

The expression in (5.16) is a distributional identity in the interior of the ordered time domain. Formula (5.17) will also specify its smooth extension to the faces.

5.3 Contraction and the positive normalizer

The forms are now compatible on all tuple simplices. To normalize them to a connection, we must show that their sum has positive contraction with the translation vector V. The choice of the lag derivative in (5.13) converts this contraction into a difference between a pairing at one time and its average at the next scale.

Write

These are the concentration quantities that will telescope.

5.4 Differential bounds and the resulting connection

Positivity now allows the normalization. Before applying the trace identity, we also need growth bounds that justify summation over all tuples. More precise bounds on the individual channel differentials will provide the input for the curvature estimates of the next section.

We now specialize the parameters of the preceding construction. At the shortest time scales we use the early channels, whose derivative bound is independent of the spatial radius. The later channels keep that radius fixed; this also fixes the terminal multiplier W while the time scales continue to increase. Their longer time scales will absorb the polynomial factor r^{10} in the later derivative bound.

Choose once and for all an integer J_* with (3/2)^{J_*-1}>200. For sufficiently large R retain the spatial radii, accuracies, and multiplier from (5.4), (5.5), and (5.9), and set

Choose b_* to be the smallest integer such that

For each even integer n>10\beta_{b_*}, use

The ratio \beta_{j+1}/\beta_j is at least two when R is sufficiently large, so these scales satisfy (5.1). Thus the construction and positivity proved above apply. For each such n, the same actual scales \beta_j/n are used in every dimension k\leq n; face restriction never changes the denominator to k.

The stage count, spatial radii, accuracies, multiplier, and numbers \beta_j now depend only on R and the fixed choices. Since \log\beta_j=(3/2)^{j-1}\log R and \log(W^2R^{100})=2C_*r+100\log R, with r a fixed multiple of R,

The particular exponents in (5.27) and (5.28) will enter the curvature bounds. They leave room for the polynomial losses in the channel derivatives.

Choose a sufficiently large fixed constant C and set

As before, l_i=1+d(h_{i-1},h_i), with periodic indexing, and \partial_i=\partial/\partial t_i.

The construction separates two estimates that the trace argument will use differently. The bounds in Proposition 5.4 justify the weighted simplex identities, while the scale-dependent bound (5.32) retains the information needed to make high-dimensional curvature Pfaffians small. We turn to those sharper estimates next.

6 Curvature estimates and the Pfaffian bound

We estimate the curvature of the connection constructed in Proposition 5.4. The smooth stages are controlled by operators on weighted Hilbert spaces. The raw stage is instead controlled by the number of nearby simplex coordinates. These two estimates will be combined only after averaging over the coordinate subsets in the Pfaffian expansion.

Throughout this section, the word metric and the fixed construction constants are understood. The constant C may depend on these data but not on R,n,g, or the tuple. A constant C_R may additionally depend on R. All parameters r,W,b_*,\beta_j,\delta_j, and a_j retain their definitions from Section 5. In particular, \delta_j=\beta_j/n and 10\delta_{b_*}<1.

We work in the section t_0=0, on the simplex

Expectation with respect to uniform simplex probability will be denoted by \mathbb E_{\Delta_n}; its density in these coordinates is n!. If Q is a two-form, its matrix is defined by Q=\sum_{i<i'}Q_{ii'}\,dt_i\wedge dt_{i'}. Our convention is

Equation (3.4) identifies the expectation below with the normalized integral in Proposition 3.3. For a subset I\subset\{1,\ldots,n\} of even cardinality, \Pf_I(Q) means the Pfaffian of the principal matrix on I, listed in increasing order. We set \Pf_{\varnothing}(Q)=1.

We prove the theorem in three steps: estimates for principal Pfaffians of the smooth stages, an averaged estimate for the raw stage, and the expansion of the normalized curvature.

6.1 Smooth stages as weighted operators

Fix a stage j>0 and a point of the simplex. Let \Lambda_j be its label set and equip (0,1)\times\Lambda_j with the measure

We use the real Hilbert space \mathcal H_j=L^2(\vartheta_j). Recall that the total mass of Y_j(u) is at most two, and is at most one at the last stage. Define the operator T_j by the kernel

where the action of g on labels is the one used in the channel construction. Oddness of U_j, symmetry of K_j, and simultaneous equivariance of K_j show that this kernel is skew. Indeed, exchange (u,y) with (v,z) and replace a by -a in the sum.

For 1\le i\le n, put

and give X_i the value zero at the remaining points. The derivatives of the zero weights vanish, and (5.32) gives

The exterior-derivative identity (5.38) is

The integral over a period can be taken with fixed endpoints because its integrand is periodic. Reducing v to (0,1) gives (6.3); the two antisymmetric terms then agree. Consequently,

Here is the finite-dimensional estimate we shall apply to this pairing. Let I have even size k>0, let E be the span of the vectors X_i, i\in I, and let s_1\ge s_2\ge\cdots be the singular values of the compression P_ET_j|_E, completed by zeros. Then

To see this, the matrix in (6.5) has rank less than k if \dim E<k, so its Pfaffian is zero. Otherwise write the X_i as columns of a square matrix V in an orthonormal basis of E. The curvature matrix is V^{\mathsf T}T_EV. Taking determinants and using \Pf(A)^2=\det A for a skew matrix gives the right side of (6.6), after Hadamard’s inequality |\det V|\le\prod_i\|X_i\|.

For 1\le j<b_*, the support of U_j is contained in [-2\delta_{j+1},2\delta_{j+1}]. At most one integer a in (6.3) contributes for a fixed pair (u,v), because 4\delta_{j+1}<1. Thus the kernel is bounded by a fixed constant and vanishes unless the circular distance between u and v is at most 2\delta_{j+1}. The mass bound on the channels implies

Bessel’s inequality gives the same upper bound for the squared Hilbert–Schmidt norm of every compression. Applying the arithmetic mean inequality to s_1^2,\ldots,s_k^2 gives

Combining this with (6.4) and (6.6), and using \beta_{j+1}=\beta_j^{3/2}, proves

In fact, the powers of the time scales simplify as

6.2 The terminal sign kernel

The last lag kernel is not supported at a small time scale. Its useful property is instead that an order comparison is constant between two disjoint ordered intervals. The following elementary approximation works even when the source and target measures differ.

For an even principal set I of size k>0, insert (6.8) in (6.6). Multiplication of the two-form by W multiplies its size-k Pfaffian by W^{k/2}. We obtain

For k\ge1, integration of \log x gives \log(k!)\ge\int_1^k\log x\,dx\ge k\log k-k, and hence k!\ge(k/e)^k. We conclude that

The estimates (6.7) and (6.10) control every smooth stage at every point of the simplex. It remains to control the raw stage in the average required for the full Pfaffian.

6.3 The raw stage and nearby coordinates

Recall J_i=\delta_{h_{i-1}}-\delta_{h_i}. The coefficient of dt_i in B_0 is

For distinct i,i' in the interior of the simplex, differentiating at the boundary t_{i'} and antisymmetrizing gives

There is at most one nonzero summand. Since K_0(h,h')=\chi_R(d(h,h')) and \chi_R is 1/R-Lipschitz,

The support of U_0 also shows that this matrix entry is zero if the circular distance between t_i and t_{i'} exceeds 2R/n.

For I\subset\{1,\ldots,n\} and i\in I, let N_i(I) be the number of indices in I, including i itself, whose times have circular distance at most 2R/n from t_i. If k=|I| is even, divide row and column i of the principal curvature matrix by l_i. Each resulting row has Euclidean norm at most (C/R)\sqrt{N_i(I)}. Hadamard’s inequality, followed by the Pfaffian–determinant identity, yields

We next average the last product. The needed assertion is an average over subsets of a fixed size; it is not a bound asserted separately for each fixed subset of the ordered coordinates.

Combining (6.11) with Lemma 6.4 gives a factor of size (CR^{-1/4})^k, once the length factors have been extracted and the coordinate subset has been averaged. The remaining task is to combine this subset average with the pointwise estimates for the smooth stages. We first make their scale dependence uniform.

6.4 Expansion of the normalized curvature

First record the dependence of the preceding estimates on R. Since r is a fixed multiple of R, the stopping condition compares

Taking j of order 1+\log R makes the first expression at least the second. As in (5.30), minimality therefore gives

For j<J_*, the constants a_j are bounded independently of R, because this is a fixed finite set of stages. Their bases in (6.7) are at most CR^{-1/8}. At the later stages,

so the intermediate bases are at most CR^{-15}(1+\log R)^{20}. At the last stage, the stopping condition \beta_{b_*}\ge W^2R^{100} gives

Together with the raw factor just obtained, all these bases are at most

for sufficiently large R, after choosing a fixed C_1 large enough.

The normalization of the connection gives

The lower bound C_{\rm nor}\ge c_0>0 from Proposition 5.4 changes the bases in (6.14) only by a fixed factor; enlarge C_1 accordingly. The last term in (6.15) is decomposable, so Q\wedge Q=0. Moreover, (5.33) implies

In particular, this stage can contribute to a Pfaffian partition only on zero or two indices. The constant C_R in (6.16) is independent of n.

There are S=b_*+2 stages in (6.15): the raw stage, the b_* smooth stages, and Q. Denote their matrices by M_0,\ldots,M_{S-1}, with M_{S-1}=Q. The Pfaffian expansion is

where all |I_j| are even and each sign \sigma is 1 or -1. There is no multiplicity factor for a fixed ordered partition. To check this normalization, expand the divided exterior power (\sum_jM_j)^m/m! as \sum_{\sum m_j=m}\prod_j M_j^{m_j}/m_j!, and then extract the coordinate volume coefficient using (6.1) on each coordinate subset.

Take absolute values in (6.17). In each term the length factors multiply to \prod_{i=1}^nl_i, independently of the partition. Fix now the even sizes k_j=|I_j|. For any given raw subset I_0 of size k_0, the number of completions to a partition of these sizes is

which is independent of I_0. Every factor other than the raw neighbor counts has already been bounded independently of the simplex point and of the actual subset. Thus, upon summing over these partitions and integrating over the simplex, the count product in (6.11) is averaged uniformly over all raw subsets of size k_0. Lemma 6.4 applies without any assertion about an individual fixed subset.

The smooth estimates also contain factors (n/k_j)^{k_j/4} or (n/k_j)^{k_j/2}. Their product is at most

For completeness, put p_j=k_j/n on the positive sizes. Concavity of the logarithm gives

exponentiating n/2 times this inequality proves (6.18). Factors for the raw and decomposable stages may be inserted here because they are at least one.

Let q=k_{S-1}. A nonzero term has q=0 or q=2. The small bases in the other stages give \varepsilon_R^{n-q}, while the decomposable stage costs at most C_R. Since 0<\varepsilon_R\le1, both possibilities are bounded by C_R\varepsilon_R^n after absorbing the single factor \varepsilon_R^{-2} into C_R. This absorption introduces no dependence on n.

For fixed sizes, the number of ordered partitions is n!/\prod_j k_j!. Summed over all admissible sizes this is at most S^n, the number of all assignments of the n indices to S stages. Combining this count, (6.18), and the preceding uniform subset average proves

Substitution of (6.14) proves (6.2), and (6.13) supplies the assertion about S. This completes the proof of Theorem 6.1.

7 Proof of the trace-support theorem

We now sum the curvature estimate against the cyclic tuple weights. The order of the two limiting choices is essential: the spatial radius is fixed first, uniformly in the conjugacy class, and only then does the simplex dimension tend to infinity.