The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

1 Introduction

The rank of a projective module over a group ring contains information about individual conjugacy classes. This refinement of ordinary rank is a basic ingredient of group-ring Euler characteristics: alternating sums of projective-module ranks retain information that a numerical Euler characteristic forgets [3, 4]. The Bass trace conjecture asks how much of that information can occur at elements of infinite order. We first specify the complex version of this question.

For a unital complex algebra A, let K_0(A) be the Grothendieck group of finitely generated projective right A-modules, and set

where [A,A] is the complex linear span of commutators. An idempotent e=(e_{ij})\in M_q(A) represents the projective right module eA^q, with matrices acting on columns by left multiplication. Its Hattori–Stallings trace is

extended additively to virtual projective classes.

Let G be a discrete group and write \con(G) for its set of conjugacy classes. The quotient \HH_0(\C G) identifies with \bigoplus_{C\in\con(G)}\C C by

Indeed, a commutator has zero sum on each conjugacy class, and the differences of conjugate group elements span the kernel of the displayed map. Such differences are commutators: tst^{-1}-s=[t,st^{-1}]. All sums arising from an element of \C G have finite support. We write \HS_G(x)(C) for the coefficient at C.

The complex group-ring Bass conjecture asks whether these coefficients vanish on every infinite-order conjugacy class, for every group and every class in K_0(\C G); see [5, Conjecture 2.1] and [11, Conjecture 2].

This resolves the complex group-ring formulation positively. Finite-order coefficients need not vanish away from the identity: for a finite group F, the idempotent |F|^{-1}\sum_{h\in F}h has coefficient |C|/|F| at every conjugacy class C. The conclusion therefore differs from the identity-only support assertion for integral group rings. That integral assertion follows in Section 9 by combining Theorem 1.1 with the known restriction on its finite-order coefficients.

1.1 Trace rank and the idempotent conjecture

For a torsion-free group the identity is the only finite-order element, so Theorem 1.1 also has numerical and ring-theoretic consequences. Define the canonical trace and the augmentation by

For a matrix b=(b_{ij})\in M_q(\C G), put \tau_{G,q}(b)=\sum_{i=1}^q\tau_G(b_{ii}), and let \epsilon_{G,q}(b)\in M_q(\C) be the entrywise augmentation. The coefficient at [1]_G of the Hattori–Stallings trace defines

The augmentation also defines the integer-valued augmentation rank

The idempotent conjecture commonly attributed to Kaplansky asks whether, for every torsion-free group G and every field k, an element e\in kG satisfying e^2=e must be 0 or 1; see [21, Problem 3]. It concerns elements of the group ring itself, not arbitrary idempotent matrices over that ring.

Part (ii) resolves the characteristic-zero Kaplansky idempotent conjecture positively, and includes coefficient domains as well as fields. Section 10 derives both parts from the trace theorem. The integer range in part (i) concerns algebraic K_0(\C G); its relation to the reduced C^*-completion is made explicit there.

The integral trace consequence also links the theorem to fixed-point theory. A map f is a homotopy idempotent if f\circ f\simeq f. Berrick, Chatterji and Mislin’s equivalences [6, Theorems 1–2] then imply that every homotopy idempotent on a connected closed smooth oriented manifold of dimension greater than two is freely homotopic to a map with exactly one fixed point. They also identify ordinary and L^2-Lefschetz numbers for pointed homotopy idempotents of connected closed smooth oriented manifolds that induce the identity on the fundamental group. Corollary 9.2 states these consequences precisely; Section 9 explains their connection to projective Euler characteristics.

1.2 The two parts of the proof

We first describe the algebraic construction to explain the geometric statement it needs. The body of the paper begins with that geometric statement, formulated independently of group rings, and completes its proof before constructing the cycles and cocycles to which it applies.

For a finitely generated group G, the algebraic construction starts with an idempotent e\in M_q(\C G) and one infinite-order element g\in G. The finitely many nonzero group coefficients of e determine a finite set of steps. Products of matrix coefficients along paths from h to gh, followed by the ordinary matrix trace, give complex weights on finite lists of group elements. Idempotency says that summing over any intermediate vertex removes that vertex. Cyclicity of matrix trace gives a second identity, a cyclic shift whose last vertex is multiplied by g.

Project the vertices to the coset set Y=\langle g\rangle\backslash G. The group H=C_G(g)/\langle g\rangle, where C_G(g) is the centralizer of g, acts freely on Y. For every positive even dimension 2m, the weighted lists define a simplicial cycle modulo this action. Its simplices use connected sets of vertices in one graph of bounded degree, fixed as m increases. A Pfaffian cocycle evaluates on this cycle as 2^{-m}\HS_G([e(\C G)^q])([g]_G). Here a cocycle means an alternating function on vertex lists that vanishes on simplicial boundaries. The explicit construction and the evaluation occupy Sections 7 and 8.

The geometric part, proved first in Sections 2–6, shows that every invariant cocycle evaluates to zero on cycles of this kind in sufficiently high dimension. We cut the supporting simplicial complex successively into pieces of bounded diameter, choosing the separating sets nearly to minimize their areas. A volume estimate and repeated slicing make the expected number of points in the final zero-dimensional set arbitrarily small. A subdivision construction turns this point count into a bound for cocycle evaluations. Equivariant chain homotopy keeps the evaluation unchanged throughout the construction.

The fixed graph degree makes increasing the dimension useful. Through each vertex there are only exponentially many connected vertex sets of a given size. On an n-simplex, Euclidean structures chosen from spanning trees of its vertex set give volume 1/n!; repeated slicing produces the same factorial in the comparison with a point of the final separating set. After these factors cancel, radii chosen in terms of the fixed degree bound make the remaining volume ratio tend to zero with n. The local replacement argument turns this strict volume margin into the small expected point count. No homological-dimension bound on H is assumed.

Two features permit this argument for arbitrary groups. A compact probability space with a free action separates overlapping translates of local modifications, without any amenability assumption. Also, each top simplex has its own averaged Euclidean area functional. The areas need not match along common faces; the successive cuts keep the relevant dimensional measures away from those faces. The cocycles need not be globally bounded: near each original simplex, the construction uses only a fixed finite set of possible vertices. Bounding the cocycle on those lists suffices for the evaluation estimate.

The final assembly in Section 9 chooses a large even dimension, forcing the individual trace coefficient to vanish. Passing from finitely generated groups to arbitrary groups uses the exact sum over subgroup conjugacy classes contained in an ambient conjugacy class. Section 10 then extracts the augmentation rank from the identity coefficient. In the scalar case this rank is 0 or 1, and the standard faithful-trace argument forces the idempotent itself to be 0 or 1. A finite-coefficient reduction extends that conclusion from complex coefficients to every commutative characteristic-zero domain.

1.3 Relation to previous work

The rank construction comes from Hattori and Stallings [16, 24]; Bass developed its relation to Euler characteristics and formulated the integral conjecture in [3]; see also [4, Conjecture (4.5)]. Bass proved complex trace vanishing at infinite-order elements for linear groups [4, Theorem (6.1)(f)]. For the integral trace, Linnell established a restriction on the possible nonzero coefficients [19, Lemma 4.1], later recovered through topological Hochschild homology by Berrick and Hesselholt [7, Theorem A]. Its finite-order consequence is the input used in Section 9.

The scalar idempotent problem also has a long history of trace methods. Formanek proved the characteristic-zero conclusion for torsion-free groups satisfying the ascending chain condition on subgroups [15, Theorem 1]. Burger and Valette revisit the theorems of Kaplansky and Zalesskii: the canonical trace of a complex scalar idempotent lies in [0,1], its endpoints force the idempotent to be trivial, and its value is rational [9, Theorems 1.3 and 2.1]. They also obtain \tau_{G,*}(K_0(\C G))\subseteq\mathbb Q [9, Corollary 1.2]. Rationality still permits nonintegral values strictly between zero and one. For torsion-free groups, Theorem 1.1 identifies the trace with augmentation rank, which excludes those values. The subsequent faithful-trace argument is classical and is included in Section 10.

Broader group classes have been treated by algebraic and analytic methods. Farrell and Linnell proved the complex Bass statement for elementary amenable groups as part of a result over integral domains [14, Theorem 1.6]. Berrick, Chatterji and Mislin obtained it for amenable groups through rational surjectivity of the degree-zero Bost assembly map for countable groups [5, Theorems 1.3–1.4 and Lemma 1.5]. Their result concerns the stronger trace assertion on \ell^1(G), with finite support on finite-order classes. An algebraic route uses induction from finite subgroups: the Farrell–Jones conjecture with complex coefficients implies the complex Bass statement [2, Theorems 1.5 and 1.9]. The present proof uses the finite support of group-ring idempotents throughout; it requires neither an assembly hypothesis nor a statement about a completed group algebra.

For some groups, analytic methods give a stronger idempotent conclusion in the reduced group C^*-algebra C_r^*(G), the operator-norm closure of \C G acting on \ell^2(G). Using Lafforgue’s Banach KK-theory, Mineyev and Yu proved this for torsion-free subgroups of hyperbolic groups [20, Theorems 19–21]. Puschnigg independently obtained the torsion-free hyperbolic case through local cyclic homology [23]; see also [20, Section 6]. Our conclusion for arbitrary torsion-free groups concerns the group ring and the trace on its algebraic K_0, as distinguished in Section 10.

The higher-dimensional algebraic construction has antecedents in the cyclic-homological approach to the conjecture. Connes constructed a pairing of even cyclic cocycles with projective classes [12, Part II, §2, Proposition 14]. Burghelea’s decomposition of cyclic homology by conjugacy classes identifies the infinite-order components through the reduced centralizers C_G(g)/\langle g\rangle [10, Proposition 1.8 and Theorem I]. Eckmann used higher-degree vanishing to obtain trace vanishing over rational group rings for specified classes of groups of finite rational homological dimension [13, §3, Theorems 3.1 and 3.3]. Sections 7–8 give direct finite-chain constructions in this tradition, including the normalization of the Pfaffian detector; no cyclic-homology theorem is needed as an input. The sparse-chain theorem replaces a homological-dimension hypothesis with connected-support control on the particular cycles.

The geometric proof develops the separating-set method of Papasoglu [22, §2, especially Lemma 2.5], the separating filtrations and simplex-counting argument of Alpert [1, Lemmas 4, 6–7], and the Cantor-parameter approach of Braun and Sauer [8, §§2, 4–5]. Alpert’s equivariant formulations [1, §4, especially Lemmas 11–12] provide the closest model for the local comparisons. Alpert’s filtration results concern manifold geometry and compatible metrics. We give the full piecewise-linear argument needed for simplex-dependent areas and arbitrary cycles in the coinvariant chain complex. The separate Pfaffian calculation converts this geometric vanishing into a statement about each conjugacy-class coefficient.

2 A vanishing theorem for sparse simplicial cycles

We formulate the geometric statement independently of group rings. Its input is a graph with a free group action and a simplicial complex whose top-dimensional vertex sets are connected in that graph. Its output is the vanishing of every invariant cocycle evaluation in sufficiently high dimension.

For a set Y, let \C^{(Y)} be the vector space with basis \{\delta_y:y\in Y\}, consisting of finite linear combinations. The augmented chains of the full simplex on Y are

The differential is contraction with the functional \alpha(\sum_y c_y\delta_y)=\sum_y c_y:

In degree zero this is the augmentation to \C; the empty wedge is 1. For a simplicial subcomplex K, its augmented oriented chains C_*(K) form a subcomplex of E_*.

If a group H acts on Y, it acts on both chain complexes by permuting vertices, with the signs prescribed by the exterior product. For an H-module V, its coinvariants are

The differentials induce differentials on coinvariants. A cycle in C_n(K)_H is a class whose boundary is zero there; an arbitrary finite chain representing it need not have zero boundary before passage to coinvariants. An H-invariant linear functional \psi:E_n\to\C descends to (E_n)_H. We call it a cocycle when \psi\partial=0 on E_{n+1}.

The bound n_0(D) depends only on the graph degree, not on H, K or \psi. There is no boundedness hypothesis on \psi. We will bound it only on a finite set of basis wedges associated with each fixed original simplex. This distinction is necessary in the application.

For the proof assume K\ne\varnothing; the empty case has no positive degree chains. Its realization |K| consists of finitely supported probability vectors p:Y\to[0,1] whose supports are simplices of K. A simplex will also denote its closed affine realization; \sigma^\circ denotes its relative interior. We call simplices of K original to distinguish them from the later subdivisions, and call its n-simplices top simplices.

The proof has three steps. First, we choose a metric and simplexwise areas for which balls have small volume in high dimension. Second, we construct separating filtrations with very few points in their final stratum, on average over a free probability parameter. Third, we use their subdivisions to construct equivariant chain maps homotopic to inclusion whose cocycle evaluations approach zero. Each approximation uses a finite filtration; no limiting filtration is needed.

3 A metric and volume bound for connected supports

We work with the graph Y, group H, and nonempty n-dimensional complex K of Theorem 2.1, where n\geq1. We need two kinds of geometric control: a metric with finite neighborhoods in which to make cuts, and Euclidean area functionals for which those cuts satisfy a slicing inequality. We will use a scaled transportation metric on |K| and, separately inside each top simplex, Euclidean structures determined by its spanning trees. Each top simplex will have volume 1/n!, while every affine branch of a distance function will have gradient norm at most one in each of these structures. Counting the possible connected vertex supports will then give the ball-volume estimate used in the separating argument.

3.1 The metric and its finite local geometry

Represent a point of |K| by a probability vector p on Y whose support lies in a simplex of K. Let d_Y denote graph distance and define

This is the scaled Kantorovich–Rubinstein metric on finitely supported probability measures, in its Lipschitz-potential form [18, pp. 1381–1382]. We establish its needed properties directly. Only finitely many terms occur in the pairing. To see that the supremum is finite and describe the distance function on each simplex, let V\subset Y be a nonempty finite set and put

This is a nonempty compact polytope. Indeed, its difference constraints and zero mean bound every coordinate by \max_{v,w\in V}d_Y(v,w). Each \ell\in\mathcal P_V extends to a 1-Lipschitz function on Y by

The difference constraints show that this extension agrees with \ell on V, and the triangle inequality proves that it is 1-Lipschitz. Conversely, a global 1-Lipschitz function restricts, after subtraction of its mean, to an element of \mathcal P_V. Constants pair to zero with p-q. Consequently, when V contains both supports, the supremum in (3.1) is the maximum over \mathcal P_V.

Symmetry follows by replacing \ell with -\ell, and the triangle inequality follows by applying each potential to p-q=(p-u)+(u-q). The indicator of a single vertex is 1-Lipschitz for d_Y, so

Thus \rho separates points and is a metric. It is H-invariant because graph automorphisms permute the admissible potentials. On probability vectors supported in a fixed finite set, it is the restriction of a norm on the finite-dimensional space of vectors with coordinate sum zero. It therefore gives the usual topology on every finite subcomplex.

For c\in|K|, write

On a fixed closed simplex \sigma, the function f_c is a maximum of finitely many affine functions: use the extreme points of \mathcal P_{V(\sigma)\cup\supp c}. Here V(\sigma) denotes the vertex set of \sigma. Taking all these extreme points makes the collection of affine branches equivariant under H.

We will also use the following consequence. For fixed c and s, only finitely many h\in H satisfy

Indeed, an intersection gives \rho(hc,c)\leq2s, and then (3.3) puts some vertex of h\supp c within distance 2s\sqrt n of \supp c. The same injectivity argument applies.

3.2 Simplex areas and distance gradients

The metric now provides finite neighborhoods in which we can make cuts. For area comparisons, we use Euclidean structures separately inside the top simplices. They need not agree on shared faces.

Fix a top simplex \sigma. For each spanning tree T of the connected graph on V(\sigma), equip its affine span with the Euclidean structure in which the tree-edge vectors \delta_v-\delta_w, with either orientation for each edge, form an orthonormal basis of the translation space. For 0\leq j\leq n, let a_j^\sigma be the average, over these finitely many trees, of Euclidean j-area on polyhedral subsets of \sigma^\circ of dimension at most j. We allow relatively open faces in these sets; pieces of dimension less than j have zero j-area, and a_0^\sigma is counting measure. These area functionals are covariant under H, since the action permutes the spanning trees.

3.3 The volume margin in high dimension

We next bound the total top-dimensional area of a metric ball by counting its possible connected vertex supports. Fix

For a specified vertex v, there are at most D^{2n} connected sets of n+1 vertices containing v. To prove this, choose a spanning tree of each such set rooted at v. A traversal using each tree edge once in each direction is a walk of length 2n, starting at v, whose visited set is exactly the prescribed set. Choosing one such walk per set gives an injection into the length-2n walks from v, of which there are at most D^{2n}.

Set k=\lfloor R\sqrt n\rfloor. A graph ball of radius k has at most

vertices, since D\geq2. As |\supp c|\leq n+1, there are at most (n+1)D^{k+1} vertices within distance k of \supp c. By (3.3), each top simplex meeting L_R(c) contains one of them. The connected-set count and Lemma 3.2 therefore give, for every c\in|K|,

Counting a simplex more than once only enlarges this upper bound.

With D fixed, the ratio of this estimate to (R-r)^n/n! is

Indeed, its logarithm is -n\log2+4D^2(\log D)\sqrt n+O_D(\log n). We may therefore choose n_0(D)\geq1 so that, for every n\geq n_0(D),

For the remainder of the geometric proof, fix such an n. The graph degree bound D, and hence R and r, remain fixed as dimensions increase. The strict margin (3.6) will convert successive area comparisons into a bound on the number of points at the bottom of a separating filtration.

4 Equivariant separating filtrations

We retain the complex K, the metric \rho, and the radii r<R from Section 3. We will repeatedly cut K into pieces of diameter at most 2R, lowering the dimension of the separating set at each step. The cuts vary over a compact probability space on which H acts freely. This allows local modifications to be separated after taking account of the parameter, even when their translates overlap in |K|.

4.1 A free probability parameter

The following construction is a finite-color variant of the coinduction argument in Braun and Sauer [8, Theorem 2.1], who credit the existence theorem to Hjorth and Molberg [17, Theorem 0.1]. We include the construction, including its torsion case, so that no free-action theorem is an input.

Fix such (X,\mu) for the rest of the geometric argument. In particular, no amenability assumption on H is needed.

4.2 The allowed filtrations

A triangulation refining K means a finite affine simplicial subdivision of each closed original simplex, with agreement on shared faces. A family T(x) of such triangulations, together with specified subcomplexes, is locally constant if, on each original simplex, all its exact affine simplices and subcomplex markings are constant on the pieces of a finite clopen partition of X. The partition may depend on the original simplex. We require equivariance: T(hx)=hT(x), and likewise for every specified subcomplex. Refinement of triangulations always preserves their earlier subcomplexes.

Begin with T_n(x)=K and Z_n(x)=|K|. At stage d, where n\ge d\ge1, suppose that P(x)=Z_d(x) is a subcomplex of T_d(x). An admissible next choice consists of a locally constant equivariant refinement T_{d-1}(x) and its subcomplex Z_{d-1}(x) such that

1. (i)

   Z_{d-1}(x)\subseteq P(x);

2. (ii)

   for every simplex F of T_d(x) contained in P(x),

   \dim\bigl(F\cap Z_{d-1}(x)\bigr)\le\dim F-1;
3. (iii)

   every connected component of P(x)\setminus Z_{d-1}(x) has \rho-diameter at most 2R.

A negative dimension bound means that the intersection is empty. Thus the new separator avoids the vertices of T_d and meets each higher-dimensional face in lower dimension; its own vertices belong to the finer triangulation T_{d-1}. Connected components use the subspace topology given by \rho. We call a filtration satisfying these conditions admissible. In particular, these conditions imply \dim Z_i(x)\le i.

The local operation behind the construction is easiest to describe at a fixed parameter. For a separator Z\subseteq P, a center c\in|K|, and t\in(r,R), remove the part of Z inside the ball and insert the corresponding slice of P:

The inserted slice prevents a component of P\setminus Z^{\prime} from passing between the ball and its exterior. Inside the ball its diameter is at most 2t; outside, the old separator remains. The next lemma makes this operation compatible with triangulations, group translates, and clopen parameter choices. It also constructs the level cuts used to start a filtration. Only the affine descriptions of distance levels enter the construction; no Euclidean metrics need to agree across faces.

The genericity against S is useful when S already contains the old separator Z: intersections of an inserted level with its faces have one less dimension. The later area comparison will use precisely this conclusion. The lemma itself does not require disjoint active balls; disjointness will be imposed when their areas are counted.

4.3 Dimension drop along the entire filtration

We record the consequence that will control chains in Section 6. For any simplex F of T_i(x) contained in Z_i(x) and any 0\le j\le i,

Indeed, the first step is the defining condition. Its intersection is a subcomplex in the next refinement. Applying the next dimension drop to each simplex of that subcomplex decreases its maximal dimension by one. Induction gives (4.2). Applied with i=n, this also shows that a proper original face meets Z_j(x) in dimension strictly less than j.

We have obtained a nonempty class of equivariant separating filtrations, together with a local cutting operation that stays within that class. The next section chooses separators with area close to the infimum and compares them with these cuts.

5 Local replacement and the number of zero strata

We now choose the separating filtration so that its zero-dimensional set has small expected cardinality in each original simplex. Throughout this section the graph, complex, parameter space, dimension, and radii are fixed as in Sections 3 and 4. In particular, the strict inequality (3.6) holds. Only the filtration will depend on a positive number \epsilon.

5.1 Successive near minimizers

The comparison with a cut through a ball follows the separating-set method of Papasoglu [22, Section 2, especially Lemma 2.5] and its iterated and equivariant forms in Alpert [1, Lemmas 6–7 and 11–12]. We prove the comparisons for our simplex-dependent areas, retaining the error from each near minimizer.

6 From separating filtrations to finite chains

Fix the dimension n and radii chosen in Section 3. For each \epsilon>0, use the filtration supplied by Proposition 5.1. We will turn it into an H-equivariant chain map

that is equivariantly chain homotopic to inclusion. Its evaluation by a fixed invariant cocycle will tend to zero. These two properties prove Theorem 2.1. Throughout this section n remains fixed while \epsilon tends to zero.

6.1 Component averages and their finite support

For a fixed parameter x, take the barycentric subdivision of the final triangulation T_0(x). Its vertices are the barycenters b_F of nonempty simplices F of T_0(x). Define

All Z_i(x) are subcomplexes of T_0(x), so b_F\in Z_i(x) is equivalent to F\subseteq Z_i(x). Let U_F be the connected component of Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x) containing b_F. For any such component U, set

The next lemma proves that J_U is finite and controls these labels as x varies. The intended image of an oriented barycentric simplex is

This choice uses the separating filtration in a specific way. If nested faces F\subset F' have the same level i, the segment joining their barycenters lies in Z_i\setminus Z_{i-1}. Except at its endpoint b_F, it lies in the relative interior of F', which the subcomplex Z_{i-1} misses; the endpoint also avoids Z_{i-1} by its level. The two barycenters therefore receive the same component vector, making any exterior term containing both zero. Along a complete flag F_0\subset\cdots\subset F_n, with \dim F_i=i, the levels are nondecreasing: if F\subset F'\subseteq Z_i, then F\subseteq Z_i. A nonzero image must consequently use all levels 0,1,\ldots,n, beginning at a point of Z_0. We will bound the number of these flags after establishing that the labels define finite chains.

Subdivide each oriented original simplex into its fundamental subdivision chain and apply the exterior assignment (6.2) to every term. This defines a map f_x:C_*(K)\to E_*. Both operations commute with the augmented boundary. For subdivision, interior facets cancel in pairs and the boundary facets give the compatible subdivision on each original face. For the second operation, each assigned vector has coefficient sum one, so contraction with the coefficient-sum functional gives

Thus f_x is an augmented chain map.

By Lemma 6.1, its value on any original simplex is constant on a finite clopen partition and is supported in a fixed finite exterior space. We may therefore define

This integral is an actual finite chain: for every c it is a finite linear combination of finitely many chain values. It commutes with the augmented boundary. Moreover, f_{hx}(hc)=h f_x(c), so invariance of \mu shows that F^\epsilon is H-equivariant.

6.2 Counting the surviving flags

We now bound the evaluation of f_x on a top simplex. The number of subdivision simplices may depend on \epsilon; the following bound depends instead on the number of zero-stratum points. The component-labeling and flag-counting method is modeled on Alpert [1, Lemmas 4 and 10]; the carrier argument below supplies the bound without a manifold hypothesis.

Figure 1 illustrates the two choices at each of the two stages when n=2.

Let \psi:E_n\to\C be the invariant cocycle in Theorem 2.1. For an original top simplex \sigma, put

This is finite, even though \psi need not be bounded on all basis simplices of the infinite vertex set. Every vector in (6.1) is a nonnegative convex combination of vertices in W_\sigma. Multilinearity therefore bounds the evaluation of each surviving term by M_\sigma. The preceding lemma and Proposition 5.1 give

All constants here are independent of \epsilon. Since chains have finite support, the same convergence holds on every fixed c\in C_n(K).

6.3 Equivariant comparison and passage to coinvariants

It remains to show that the small evaluations just obtained equal the original cocycle evaluation on every coinvariant cycle. We do this by constructing a homotopy before taking coinvariants.

7 Cycles from an idempotent

We now return to group rings. For an idempotent and one infinite-order conjugacy class, we will construct cycles in every positive even degree. Their vertices follow steps in one fixed graph of bounded degree. In sufficiently high dimension, the supporting complexes will therefore satisfy the hypotheses of Theorem 2.1. We first record why it suffices to perform this construction for finitely generated groups.

The use of cyclic idempotent expressions and reduced centralizers has its conceptual background in Connes’s pairing with projective classes [12, Part II, Section 2, Proposition 14] and Burghelea’s conjugacy-class decomposition [10, Theorem I]. Our finite path identities and the detector’s normalization are proved directly; no cyclic-homology identification is assumed.

7.1 A fixed graph and finite path sums

For the rest of this section, fix a finitely generated group G, an idempotent e\in M_r(\C G), and an infinite-order element g\in G. Write

The set S_0 is finite. Idempotence says that \sum_{uv=t}A_uA_v=A_t for every t\in G. Thus the matrix-valued kernel

is invariant under simultaneous left translation and satisfies

All these sums are finite. In particular, the order of the matrix factors agrees with the column convention for right modules.

Put

Choose a finite symmetric generating set S\subset G\setminus\{1\} containing S_0\setminus\{1\} and its inverses. Give Y the simple undirected graph whose edges join \bar h to \overline{hs} for s\in S; loops are omitted and repeated edges identified. The graph is connected and has degree at most the fixed integer

The group H acts by left multiplication. This is well-defined because J centralizes g, and it preserves right-step edges. It is free on vertices: \overline{ch}=\bar h with c\in J implies ch=g^q h for an integer q, and hence c=g^q. Both H and Y are countable. We keep this graph and this value of D throughout all subsequent choices of dimension.

For k\ge1, let Q_k=J\backslash G^k, with simultaneous left translation on tuples. Define the weight of a tuple by

where w_1(h_0)=\tr(a(h_0,gh_0)). These weights are J-invariant. They define a finite complex linear combination v_k of elements of Q_k. Indeed, a nonzero weight has increments

in S_0, with

An increment list determines the tuple from h_0. Any two choices of h_0 with this conjugate differ by left multiplication in J, so the list determines at most one element of Q_k. Hence v_k has at most |S_0|^k nonzero coefficients. For a J-invariant function p on G^k, write v_k(p) for the finite weighted sum of its values. These weights need not be real or positive.

7.2 Even-dimensional cycles

The path identities now turn the finite sums into cycles. We use the exterior complex E_* on the graph Y. For k\ge1, define

The map is well-defined because changing a tuple by an element of J changes its projected vertices by a simultaneous H-translation. Set b_k=\pi_k(v_k), and, for n=2m\ge2, set

We have obtained the required sparse cycles in arbitrarily large even degrees. The underlying path sums all have total weight \lambda, the trace coefficient in (7.4). The next section constructs a cocycle whose value on z_{2m} recovers a nonzero scalar multiple of that coefficient.

8 A Pfaffian cocycle detecting the trace coefficient

Retain the finitely generated group G, idempotent, infinite-order element g, and notation J,H,Y,v_k,\lambda,z_{2m} of Section 7. We construct an H-invariant cocycle whose value on z_{2m} is 2^{-m}\lambda. The construction first records the relative powers of g between two lifts of vertices of Y. An alternating Pfaffian expression will then remove the dependence on those lifts.

8.1 Relative exponents between lifts

The function \beta itself need not descend to Y\times Y. In fact, (8.1) gives \beta(h,g^k h)=k even though these two lifts project to the same vertex. Nor is there a difficulty if an element \bar c\in H exchanges two distinct projected vertices. If a lift c\in J satisfies c h_0=g^a h_1 and c h_1=g^b h_0, its invariance merely requires

which is precisely what antisymmetrization gives. Real values are useful here: for G=\langle t\rangle and g=t^2, the value \beta(1,t)=1/2 is forced.

8.2 The cyclic Pfaffian identity

In degree two, a lift-independent expression is

Changing h_i to g^{r_i}h_i adds (r_2-r_1)-(r_2-r_0)+(r_1-r_0)=0, so c depends only on the projected vertices. In higher even degrees we replace \beta by a Pfaffian and take the same alternating deletion sum. The Pfaffian still depends on the lifts; its deletion sum will define the cocycle. We first compute the weighted Pfaffian sum, then prove the descent and cocycle properties of its deletion sum.

For a list \mathbf h=(h_0,\ldots,h_{2m-1}) with m\ge1, let

We use the convention that p_m(\mathbf h) is the coefficient of \varepsilon_0\wedge\cdots\wedge\varepsilon_{2m-1} in b^m/m!, where \varepsilon_i are the coordinate one-forms on the space of list positions and

In particular, p_1(h_0,h_1)=\beta(h_0,h_1), and p_m is alternating in its entries and J-invariant. The finite weighted sums

are therefore defined. No boundedness of p_m on all lists is needed.

8.3 Descent to an exterior cocycle

The preceding calculation detects the coefficient on lists of lifts. We next descend it to the exterior complex on Y, where the sparse-chain theorem applies.

9 Completion of the trace theorem

We now combine the geometric vanishing with the detector. The order of the choices matters: the graph is fixed by the idempotent before its cycle dimension is chosen.

9.1 The integral trace

For the integral group ring, the same idempotent formula defines

where commutators are now taken with their additive span. The integral Bass trace conjecture asks for support only at the identity class.

9.2 Homotopy idempotents and Lefschetz numbers

The integral support statement also has fixed-point consequences. A self-map f is a homotopy idempotent if f\circ f\simeq f. Such a map on a connected finite CW complex X admits maps u:X\to Y and v:Y\to X with

[6, Section 3]. The second identity says that Y is finitely dominated by X. For a pointed homotopy idempotent with f_\#=\id_G, where G=\pi_1(X), the induced maps u_\# and v_\# identify \pi_1(Y) with G. The universal-cover chains of Y are chain homotopy equivalent to a finite complex P_* of finitely generated projective \Z G-modules. The sum \sum_i(-1)^i[P_i]\in K_0(\Z G) is its projective Euler class. The Hattori–Stallings trace of this class records the fixed-point class indices of f: it is the Reidemeister trace, with conjugacy classes inverted when passing from the left-module convention of [6, Lemma 4.1] to ours. The augmentation trace and the identity coefficient of this projective class give the ordinary and L^2-Lefschetz numbers, respectively [6, Lemmas 8.1–8.2]. Berrick, Chatterji and Mislin use these relations to deduce the following consequences of the integral support statement and its numerical trace consequence.

10 Torsion-free groups: trace rank and idempotents

For a torsion-free group, the support theorem leaves only the identity conjugacy class. We first identify its coefficient with augmentation rank, and then use the two possible scalar ranks to determine scalar idempotents.

For matrices the conclusion is numerical. For example, \operatorname{diag}(1,0)\in M_2(\C G) is an idempotent other than zero and the identity, with augmentation rank one. The scalar case is special because its augmentation rank can only be zero or one.