Rank gradient versus stable integral simplicial volume
type: Item
instance of: scholarly article
title: Rank gradient versus stable integral simplicial volume
zbMATH Open document ID: 1399.57007
author: Clara Löh
published in: Periodica Mathematica Hungarica
publication date: 2018-04-27T00:00:00Z
review text: If \(\Gamma\) is a finitely generated group and \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), then the rank gradient of \(\Gamma\) with respect to \(\Gamma_*\) is defined as \(\operatorname{rg}(\Gamma,\Gamma_*)=\inf\limits_{k\in\mathbb N}\frac{d(\Gamma_k)-1}{[\Gamma:\Gamma_k]}\), while the absolute rank gradient of \(\Gamma\) is defined as \(\operatorname{rg}(\Gamma)=\inf\limits_{H\in F(\Gamma)}\frac{d(H)-1}{[\Gamma:H]}\), where \(F(\Gamma)\) is the set of all finite index subgroups of \(\Gamma\) and \(d(\Gamma)\) is the rank of \(\Gamma\), i.e., the minimal size of a generating set of \(\Gamma\). If \(\Gamma\) is the fundamental group of an oriented closed connected \(n\)-manifold \(M\), \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), and \(M_k=\widetilde M/\Gamma_k\) is the covering manifold of \(M\) associated with the subgroup \(\Gamma_k\), then the stable integral simplicial volume of \(M\) with respect to \(\Gamma_*\) is defined as \(\| M\| ^{\Gamma_*}_{\mathbb Z}=\inf\limits_{k\in\mathbb N}\frac{\| M_k\|_{\mathbb Z}}{[\Gamma:\Gamma_k]}\), while the stable integral simplicial volume of \(M\) is defined as \(\| M\| ^{\infty}_{\mathbb Z}=\inf\limits_{H\in F(\Gamma)}\frac{\| \widetilde M\|_{\mathbb Z}}{[\Gamma:H]}\). In this paper, the author observes that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups. It is shown that if \(M\) is an oriented closed connected manifold with fundamental group \(\Gamma\) and \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), then \(\operatorname{rg}(\Gamma,\Gamma_*)\leq \| M\| ^{\Gamma_*}_{\mathbb Z}\). In particular, \(\operatorname{rg}(\Gamma)\leq \| M\| ^{\infty}\).
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 6865420
zbMATH Keywords: stable integral simplicial volume
zbMATH Keywords: rank gradient
MaRDI profile type: Publication
full work available at URL: s10998-017-0212-1
OpenAlex ID: W2607330716
cites work: Rank gradient, cost of groups and the rank versus Heegaard genus problem
cites work: The rank gradient from a combinatorial viewpoint.
cites work: Stable complexity and simplicial volume of manifolds
cites work: Integral foliated simplicial volume of aspherical manifolds
cites work: Cost of equivalence relations and groups
cites work: \(\ell^2\) invariants of equivalence relations and groups
cites work: Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
cites work: Volume and bounded cohomology
cites work: Rank gradient and cost of Artin groups and their relatives.
cites work: Topics in orbit equivalence
cites work: Rank and deficiency gradients of generalized Thompson groups of type \(F\).
cites work: Expanders, rank and graphs of groups
cites work: Integral foliated simplicial volume of hyperbolic 3-manifolds
cites work: Approximating \(L^ 2\)-invariants by their finite-dimensional analogues
cites work: Rank Gradient andp-gradient of Amalgamated Free Products and HNN Extensions
DOI: 10.1007/S10998-017-0212-1
description: scientific article; zbMATH DE number 6865420