On an isoperimetric inequality for infinite finitely generated groups
type: Item
instance of: scholarly article
title: On an isoperimetric inequality for infinite finitely generated groups
zbMATH Open document ID: 0988.20028
DOI: 10.1016/S0040-9383(99)00043-9
author: Andrzej Zuk
published in: Topology
publication date: 2002-06-16T00:00:00Z
review text: Let \(\Gamma\) be an infinite discrete group with a fixed finite symmetric set \(S=S^{-1}\) of generators. Then: \(|A|\leq\sum_{\gamma\in\partial A}\text{dist}(e,\gamma)\) where \(|A|\) denotes the number of elements in \(A\) and \(\partial A=\{\gamma\not\in A;\;s\gamma\in A,\;s\in S\}\); \(\text{dist}(e,\gamma)\) is the minimal number of generators needed to represent \(\gamma\in\Gamma\). The proof rests on the construction of a finitely additive and invariant measure on certain sets of geodesics in \(\Gamma\). Using a result of \textit{J.-C. Sikorav} [Growth of a primitive of a differential form, preprint, Toulouse (1998)] one obtains a result for the universal cover \(\widetilde M^n\) of a compact Riemannian manifold with infinite fundamental group: There is a primitive of the volume form on \(\widetilde M^n\) which has at most linear growth.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1472310
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: finitely generated groups
zbMATH Keywords: generators
zbMATH Keywords: numbers of elements
zbMATH Keywords: numbers of generators
zbMATH Keywords: measures
zbMATH Keywords: geodesics
zbMATH Keywords: universal covers
zbMATH Keywords: compact Riemannian manifolds
zbMATH Keywords: fundamental groups
reviewed by: Gerhard Burde
MaRDI profile type: Publication
Wikidata QID: Q126376956
full work available at URL: s0040-9383(99)00043-9
OpenAlex ID: W2051261060
description: scientific article; zbMATH DE number 1472310