Growth tightness for groups with contracting elements.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1316.20047
DOI: 10.1017/S0305004114000322
author: Wen-Yuan Yang
published in: Mathematical Proceedings of the Cambridge Philosophical Society
publication date: 2014-11-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.5623
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 6365833
zbMATH Keywords: relatively hyperbolic groups
zbMATH Keywords: growth tightness
zbMATH Keywords: Floyd boundaries
zbMATH Keywords: contracting elements
zbMATH Keywords: non-positive curvature
zbMATH Keywords: geodesics
zbMATH Keywords: finitely generated groups
zbMATH Keywords: growth rates
zbMATH Keywords: Cayley graphs
zbMATH Keywords: CAT(0) groups
MaRDI profile type: Publication
arXiv ID: 1301.5623
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title: Growth tightness for groups with contracting elements.
review text: A finitely generated group \(G\) is growth tight if for every quotient \(\overline G=G/\Gamma\) by an infinite normal subgroup the exponential growth rate of balls in Cayley graphs of \(\overline G\) is strictly smaller than for \(G\). The second main result of the article is that if \(G\) admits a non-trivial Floyd boundary (with respect to some generating set and Floyd function) then it is growth tight.NEWLINENEWLINE If a group acts properly on a proper geodesic metric space, it inherits a pseudometric from that action and growth tightness can be expressed with respect to this metric (taking for the quotients the metric inherited from the action on the quotient space). The first main result of the article is that if a group that is not virtually cyclic acts geometrically on a proper geodesic metric space and if the action admits a contracting element then the group is growth tight with respect to that action. The second main result is deduced from the first.NEWLINENEWLINE An element \(h\in G\) is contracting if the projection of a quasi-geodesic to the group \(\langle h\rangle\) has uniformly small diameter (depending on the distance between the quasi-geodesic and \(\langle h \rangle\)).NEWLINENEWLINE Each of these properties represents a kind of negative curvature phenomenon. For example, it follows directly from the first main theorem that hyperbolic groups are growth tight (which was known before [\textit{S. Sabourau}, J. Mod. Dyn. 7, No. 2, 269-290 (2013; Zbl 1286.20057)]) and that CAT(\(0\)) groups with a rank-\(1\) element are growth tight (using that rank-\(1\) elements are contracting by \textit{M. Bestvina} and \textit{M. Feighn} [J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029); addendum and correction ibid. 43, No. 4, 783-788 (1996; Zbl 0862.57027)]). Since relatively hyperbolic groups have non-trivial Floyd boundary by \textit{V. Gerasimov} [Geom. Funct. Anal. 22, No. 5, 1361-1399 (2012; Zbl 1276.20050)], it follows from the second main theore…
reviewed by: Stefan Witzel
description: scientific article; zbMATH DE number 6365833