The $L^2$-Brown formula $E(G)(s)=\chi^{(2)}(C_G(s))$ for groups of type FP over $\mathbb{C}$ (Chatterji–Mislin Conjecture 1) and the scope of the class $\mathcal{B}_\infty$

The complete Euler characteristic $E(G)$ of a group $G$ of type FP over $\mathbb{C}$ is the Hattori–Stallings trace of the Wall element of a finite resolution by finitely generated projective $\mathbb{C}G$-modules; it is a class function on conjugacy classes refining the ordinary Euler characteristic. Brown (1982) conjectured, and proved under finiteness conditions on centralizers, that $E(G)(s)$ equals the Euler characteristic of the centralizer $C_G(s)$ at elements $s$ of finite order and vanishes at elements of infinite order. Chatterji and Mislin (2009) proposed the $L^2$-analytic generalization, their Conjecture 1: if $G$ is of type FP over $\mathbb{C}$ and the centralizer of every finite-order element has finite $L^2$-Betti numbers, then $E(G)(s)=\chi^{(2)}(C_G(s))$ for every $s\in G$. The conjecture subsumes, at elements of infinite order, the Bass conjecture over $\mathbb{C}$ for FP groups; it is known for groups with cocompact classifying space for proper actions, for the closure classes $\mathcal{B}\subset\mathcal{B}_\infty$ built by the authors, and for all groups satisfying the Bass conjecture over $\mathbb{C}$ at infinite-order elements, but remains open for a general group of type FP over $\mathbb{C}$. The authors also pose the scope question of whether every group of type FP over $\mathbb{C}$ lies in $\mathcal{B}_\infty$. Both questions remain open (see [1]).

Background

For a group $G$, the complex group ring $\mathbb{C}G$ consists of the finitely supported complex-valued functions on $G$, with multiplication induced by the group operation. A $\mathbb{C}G$-module is finitely generated projective if it is a direct summand of a finite-rank free module; the Grothendieck group $K_0(\mathbb{C}G)$ of such modules records stable projective classes. The Hattori–Stallings trace is the map $\mathrm{HS}\colon K_0(\mathbb{C}G)\to\bigoplus_{[G]}\mathbb{C}$ on the set of conjugacy classes $[G]$ of $G$ that sends a projective module represented by an idempotent matrix $M=(m_{ij})$ over $\mathbb{C}G$ to the class function $s\mapsto\sum_i m_{ii}(s)$; it is a noncommutative analogue of the ordinary trace of a projection. Bass conjectured (1976) that for a finitely generated projective $\mathbb{Z}G$-module the Hattori–Stallings rank vanishes on every group element other than the identity; the stronger Bass conjecture over $\mathbb{C}$ asserts that for a finitely generated projective $\mathbb{C}G$-module $P$ one has $\mathrm{HS}(P)(s)=0$ for every element $s$ of infinite order. It is known for linear groups, groups of cohomological dimension at most 2 over $\mathbb{C}$, amenable groups (a group is amenable if it admits a finitely additive invariant mean), and more generally for all groups satisfying the Bost conjecture — the assertion that the Bost assembly map from the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper $G$-actions to $K_0(\ell^1G)$ is an isomorphism, where $\ell^1G$ is the Banach algebra of summable functions on $G$; this route is due to Berrick–Chatterji–Mislin (2004). The Bost conjecture itself is open for many groups, and the Bass conjecture over $\mathbb{C}$ is prominently open for general profinite groups (compact inverse limits of finite groups, viewed as abstract groups), where only modules induced from $\mathbb{Z}G$-projectives are settled by a theorem of Linnell (1983).

A group $G$ is of type FP over $\mathbb{C}$ if the trivial module $\mathbb{C}$ admits a resolution $0\to P_n\to\cdots\to P_0\to\mathbb{C}$ by finitely generated projective $\mathbb{C}G$-modules (type FF over $\mathbb{C}$ if the $P_i$ may be chosen free). The alternating sum $W(G)=\sum_i(-1)^i[P_i]\in K_0(\mathbb{C}G)$, the Wall element, depends only on $G$, and its Hattori–Stallings trace $E(G)=\sum_{[s]}E(G)(s)[s]$ is the complete Euler characteristic of $G$ (Stallings). The value $e(G)=E(G)(1)$ is the classical Euler characteristic of Bass and Chiswell. The $i$-th $L^2$-Betti number of $G$ is $\beta_i(G)=\dim_{\mathcal{N}(G)}H_i(G;\mathcal{N}(G))$, the von Neumann dimension of group homology with coefficients in the group von Neumann algebra $\mathcal{N}(G)$ (the algebra of bounded operators on $\ell^2G$ commuting with the left regular representation, equipped with its canonical trace and dimension function); if the alternating sum converges, the $L^2$-Euler characteristic is $\chi^{(2)}(G)=\sum_i(-1)^i\beta_i(G)$. A theorem of Cheeger–Gromov (1986) says that if $G$ contains an infinite normal amenable subgroup then all $\beta_i(G)$ vanish; since $s$ is central in its centralizer $C_G(s)=\{g\in G:gs=sg\}$, this gives $\chi^{(2)}(C_G(s))=0$ for every element $s$ of infinite order.

Brown (1982) conjectured that, under suitable finiteness conditions on centralizers, $E(G)(s)=e(C_G(s))$ for $s$ of finite order and $E(G)(s)=0$ otherwise, proving it in many cases, notably for groups admitting a cocompact $\underline{E}G$. His hypotheses require in particular that the relevant centralizers be of type FP over $\mathbb{C}$ — a genuine obstruction, since Leary and Nucinkis (2003) constructed groups of type FP over $\mathbb{C}$ containing finite-order elements whose centralizers are not of type FP over $\mathbb{C}$, so that Brown's formula does not even make sense for them. Chatterji and Mislin (2009) placed Brown's question in the $L^2$-setting with their Conjecture 1 below, which applies where Brown's formula cannot and specializes to Brown's formula when all centralizers are of type FP over $\mathbb{C}$; for a group $G$ with a finite $K(G,1)$ it reduces to Atiyah's theorem $\chi(G)=\chi^{(2)}(G)$. They prove Conjecture 1 for a closure class $\mathcal{B}$ of groups built from groups with cocompact $\underline{E}G$ and groups $H\times\mathbb{Z}$ via products and finite graphs of groups subject to the paper's Condition (F) (a graph of groups is a group-theoretic analogue of a graph of spaces; taking its fundamental group glues vertex groups along edge subgroups), and they prove the infinite-order case of the formula for the larger class $\mathcal{B}_\infty$ of Definition 4.1. Examples in $\mathcal{B}_\infty$ include word-hyperbolic, CAT(0), Coxeter, braid, Artin, and mapping-class groups, knot groups, one-relator and $S$-arithmetic groups, and amenable groups of type FP over $\mathbb{C}$. What is not known is whether every group of type FP over $\mathbb{C}$ satisfies the conjecture, and whether every such group already lies in $\mathcal{B}_\infty$.

Problem Statement

Conjecture 1 (Chatterji–Mislin, 2009). Let $G$ be a group of type FP over $\mathbb{C}$ such that the centralizer of every element of finite order in $G$ has finite $L^2$-Betti numbers (so that $\chi^{(2)}(C_G(s))$ is defined for every $s\in G$; at elements of infinite order all $L^2$-Betti numbers of the centralizer vanish by Cheeger–Gromov). Prove, or refute by an explicit example, that for every $s\in G$

$$E(G)(s)=\chi^{(2)}(C_G(s)),$$

where $E(G)$ is the complete Euler characteristic of $G$ (the Hattori–Stallings trace of the Wall element $W(G)=\sum_i(-1)^i[P_i]\in K_0(\mathbb{C}G)$ of a finite projective resolution) and $\chi^{(2)}$ is the $L^2$-Euler characteristic. The formula is known when $G$ admits a cocompact classifying space for proper actions (Brown's formula), for the closure class $\mathcal{B}$ and at infinite-order elements for the class $\mathcal{B}_\infty$ of Chatterji–Mislin, and at infinite-order elements for any group of type FP over $\mathbb{C}$ satisfying the Bass conjecture over $\mathbb{C}$; the general case remains open. Subsidiary scope question posed in the same paper: is every group of type FP over $\mathbb{C}$ a member of $\mathcal{B}_\infty$, the smallest class of groups containing all groups of type FF over $\mathbb{C}$, all groups of type FP over $\mathbb{C}$ satisfying the Bass conjecture over $\mathbb{C}$, all groups with cocompact $\underline{E}G$, and all groups $H\times\mathbb{Z}$ with $H$ of type FP over $\mathbb{C}$, and closed under finite products and under passing to fundamental groups of finite graphs of groups? The authors state that they know of no group of type FP over $\mathbb{C}$ outside $\mathcal{B}_\infty$; deciding this delimits exactly how far the known stability-based verifications of the conjecture reach. The two questions are logically independent: resolving the formula alone does not settle the scope question, and settling membership in $\mathcal{B}_\infty$ alone does not resolve Conjecture 1.

References

1. I. Chatterji, G. Mislin, Hattori-Stallings trace and Euler characteristics for groups, in: Geometric and Cohomological Methods in Group Theory, London Mathematical Society Lecture Note Series 358, Cambridge University Press, 2009, pp. 256–271. DOI: 10.1017/cbo9781139107099.007; arXiv:0910.4419. https://arxiv.org/abs/0910.4419 (primary source)
2. H. Bass, Euler characteristics and characters of discrete groups, Inventiones Mathematicae 35 (1976), 155–196. DOI: 10.1007/BF01390137. https://doi.org/10.1007/BF01390137
3. K. S. Brown, Complete Euler characteristics and fixed-point theory, Journal of Pure and Applied Algebra 24 (1982), 103–121. DOI: 10.1016/0022-4049(82)90008-1. https://doi.org/10.1016/0022-4049(82)90008-1
4. A. J. Berrick, I. Chatterji, G. Mislin, From acyclic groups to the Bass conjecture for amenable groups, Mathematische Annalen 329 (2004), 597–621. DOI: 10.1007/s00208-004-0521-6. https://doi.org/10.1007/s00208-004-0521-6
5. I. Leary, B. Nucinkis, Some groups of type VF, Inventiones Mathematicae 151 (2003), 135–165. DOI: 10.1007/s00222-002-0254-7. https://doi.org/10.1007/s00222-002-0254-7
6. J. Cheeger, M. Gromov, $L^2$-Cohomology and group cohomology, Topology 25 (1986), no. 2, 189–215. DOI: 10.1016/0040-9383(86)90039-X. https://doi.org/10.1016/0040-9383(86)90039-X
7. P. A. Linnell, Decomposition of augmentation ideals and relation modules, Proceedings of the London Mathematical Society (3) 47 (1983), 83–127. DOI: 10.1112/plms/s3-47.1.83. https://doi.org/10.1112/plms/s3-47.1.83
8. W. Lück, H. Reich, The Baum–Connes and the Farrell–Jones conjectures in K- and L-theory, in: Handbook of K-Theory, Vols. 1–2, Springer, Berlin, 2005, pp. 703–842. ISBN 3-540-23019-X. DOI: 10.1007/978-3-540-27855-9_15. https://doi.org/10.1007/978-3-540-27855-9_15; arXiv:math/0402405. https://arxiv.org/abs/math/0402405
9. R. Ji, C. Ogle, B. Ramsey, Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture, Journal of Noncommutative Geometry 4 (2010), no. 1, 83–124. DOI: 10.4171/jncg/50. https://doi.org/10.4171/jncg/50
10. J. H. Jo, On vanishing of $L^2$-Betti numbers for groups, Journal of the Mathematical Society of Japan 59 (2007), no. 4. DOI: 10.2969/JMSJ/05941031. https://doi.org/10.2969/JMSJ/05941031
11. J. H. Jo, Atiyah's $L^2$-index theorem, Kan–Thurston construction, and weak Bass conjecture, Boletim da Sociedade Brasileira de Matemática, Nova Série 45 (2014), no. 2, 355–369. DOI: 10.1007/s00574-014-0053-y. https://doi.org/10.1007/s00574-014-0053-y
12. J. H. Jo, Equivariant $L^2$-Euler characteristics of $G$-CW-complexes, Arkiv för Matematik 55 (2017), no. 1, 155–164. DOI: 10.4310/ARKIV.2017.v55.n1.a7. https://doi.org/10.4310/ARKIV.2017.v55.n1.a7
13. I. Chatterji, S. Hughes, P. Kropholler, The first $\ell^2$-Betti number and groups acting on trees, Proceedings of the Edinburgh Mathematical Society 64 (2021), no. 4, 916–923. DOI: 10.1017/S0013091521000663; arXiv:2012.13368. https://arxiv.org/abs/2012.13368
14. I. Chatterji, G. Mislin, On the idempotent conjecture for Sidki doubles, in: Geometric Methods in Group Theory (Papers dedicated to Ruth Charney), Séminaires et Congrès 34, Société Mathématique de France, 2025, pp. 33–46; arXiv:2204.02176. https://arxiv.org/abs/2204.02176