Cohomological lower bounds for isoperimetric functions on groups
type: Item
instance of: scholarly article
title: Cohomological lower bounds for isoperimetric functions on groups
zbMATH Open document ID: 0933.20026
DOI: 10.1016/S0040-9383(97)00070-0
published in: Topology
publication date: 2000-01-20T00:00:00Z
review text: In previous papers [see \textit{S. M. Gersten}, in Combinatorial and geometric group theory, Lond. Math. Soc. Lect. Note Ser. 224, 87-104 (1995; Zbl 0851.20036)], the author has worked out an \(\ell_\infty\)-cohomology which is adapted to the study of isoperimetric functions on finitely presented groups. If a finitely generated group \(G\) possesses an Eilenberg-MacLane space \(X\) of type \(K(G,1)\), where \(X\) is a CW-complex with a finite \(n\)-skeleton \(X^{(n)}\), then \(H_{(\infty)}^i(G)\) is defined for \(i\leq n\) by taking uniformly bounded cellular cocycles in the universal cover \(\widetilde X\), modulo the image under coboundary of uniformly bounded cochains. The groups \(H_{(\infty)}^i(G)\) are quasi-isometry invariants. A class in \(H_{(\infty)}^2(G)\) gives a lower bound for the Dehn function of a finite presentation for \(G\) (the Dehn function is the minimal isoperimetric function for the presentation). If the cohomology class is zero, this lower bound is linear. There is a Mayer-Vietoris exact sequence for \(\ell_\infty\)-cohomology which is associated to a splitting of groups. If \(G\) splits over a finitely presented subgroup, then the author constructs classes in \(H_{(\infty)}^2(G)\) which reflect the splitting and which serve as lower bounds for isoperimetric functions. He proves that \(H_{(\infty)}^2(G)=0\) if \(G\) is word hyperbolic and he presents results on the converse of this theorem. These results concern the Bestvina-Feighn combination theorem for word hyperbolic groups. The author gives several theorems on the non-vanishing of the \(\ell_\infty\)-cohomology. He introduces the notion of metabolic groups, as finitely presented groups such that \(H_{(\infty)}^2(G,A)=0\) for all normed Abelian coefficient groups \(A\). He shows that such groups \(G\) satisfy the linear isoperimetric inequality for cycles in all dimensions and that all \(\ell_\infty\)-cohomology groups for all coefficients in degrees at least 2 vanish. The corresponding …
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F06
zbMATH DE Number: 1307791
zbMATH Keywords: word hyperbolic groups
zbMATH Keywords: Dehn functions
zbMATH Keywords: isoperimetric functions
zbMATH Keywords: combination theorems
zbMATH Keywords: \(\ell_\infty\)-cohomology
zbMATH Keywords: splittings of groups
zbMATH Keywords: group actions on trees
zbMATH Keywords: Mayer-Vietoris sequences
zbMATH Keywords: finitely presented groups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: quasi-isometry invariants
zbMATH Keywords: finite presentations
zbMATH Keywords: metabolic groups
zbMATH Keywords: isoperimetric inequalities
author: Steve M. Gersten
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
Wikidata QID: Q127443798
description: scientific article; zbMATH DE number 1307791