The Atiyah conjecture and \(L^2\)-cohomology computations
type: Item
instance of: scholarly article
title: The Atiyah conjecture and \(L^2\)-cohomology computations
zbMATH Open document ID: 1496.20068
author: Kevin Schreve
author: Wiktor J. Mogilski
published in: Geometriae Dedicata
publication date: 2018-08-23T00:00:00Z
review text: Let \(G\) be a discrete group and \(X\) a proper cocompact \(G\)-\(CW\)-complex. The \textit{strong Atiyah conjecture} states that \(\operatorname{lcm}G\cdot b^{(2)}_{k}(X;G)\) is an integer, where \(\operatorname{lcm}G\) denotes the least common multiple of orders of finite subgroups of \(G\) and \(b^{(2)}_{k}(X;G)\) denotes the \(k\)th-\(\ell^{2}\) Betti number of \(X\). This paper extends the validity of the strong Atiyah conjecture to some classes of groups and spaces including certain Coxeter groups and fundamental groups of special cube complexes which are quasi-isometric to a wedge of Euclidean half spaces. A nonpositively curved complex is \textit{sparse} if it is quasi-isometric to a wedge of finitely many Euclidean half-spaces. A cube complex \(X\), admits a \textit{cubical hierarchy} if \(X\) can be cut along embedded hyperplanes to a disjoint union of compact contractible cube complexes. The main result of this paper is the following: Theorem. Let \(G\) be the fundamental group of a sparse cube complex \(X\) that admits a cubical hierarchy and is hyperbolic relative to the collection of peripheral subgroups \(\mathcal{P}\). Suppose each peripheral subgroup is free abelian. Then, a finite index subgroup of \(G\) is a retract of a virtually cocompact special group \(G\). Hence, any group commensurable to \(G\) satisfies the strong Atiyah conjecture. As a consequence of this result, the authors prove vanishing results of \(\ell^{2}\)-homology for certain Coxeter groups.
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 6924057
zbMATH Keywords: strong Atiyah conjecture
zbMATH Keywords: Coxeter groups
zbMATH Keywords: \(L^2\)-Betti numbers
zbMATH Keywords: Singer conjecture
zbMATH Keywords: special groups
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
full work available at URL: s10711-017-0293-2
OpenAlex ID: W2765393749
cites work: The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).
cites work: Q4086472
cites work: Rational group ring elements with kernels having irrational dimension.
cites work: Buildings with isolated subspaces and relatively hyperbolic Coxeter groups.
cites work: Q5431018
cites work: Some aspherical manifolds
cites work: Vanishing theorems and conjectures for the 2-homology of right-angled Coxeter groups
cites work: Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products.
cites work: \(\ell^2\)-homology of right-angled Coxeter groups based on barycentric subdivisions
cites work: Weighted \(L^2\)-cohomology of Coxeter groups.
cites work: Thin buildings.
cites work: Introduction to \(\ell_2\)-methods in topology: Reduced \(\ell_2\)-homology, harmonic chains, \(\ell_2\)-Betti numbers
cites work: Special cube complexes
cites work: Coxeter groups are virtually special
cites work: Incoherent Coxeter groups.
cites work: Division Rings and Group von Neumann Algebras
cites work: Groups with the same cohomology as their profinite completions.
cites work: \(L^ 2\)-topological invariants of 3-manifolds
cites work: Q3148984
cites work: The fattened Davis complex and weighted \(L^2\)-(co)homology of Coxeter groups
cites work: Coxeter Groups act on CAT(0) cube complexes
cites work: The \(L^2\)-(co)homology of groups with hierarchies
cites work: Q5485637
cites work: Q5485639
cites work: Q5485638
cites work: The strong Atiyah conjecture for virtually cocompact special groups
cites work: Galois Cohomology
cites work: The topology of isospectral manifolds of tridiagonal matrices
cites work: Q4898825
DOI: 10.1007/S10711-017-0293-2
description: scientific article; zbMATH DE number 6924057