Floyd maps for relatively hyperbolic groups.
type: Item
instance of: scholarly article
title: Floyd maps for relatively hyperbolic groups.
zbMATH Open document ID: 1276.20050
author: Victor Gerasimov
published in: Geometric and Functional Analysis. GAFA
publication date: 2013-01-14T00:00:00Z
full work available at URL: http://edoc.mpg.de/629650
review text: Given a finitely generated group \(G\) and a finite generating set, the edges in the Cayley graph have length \(1\). W. J. Floyd conformally deformed the metric to get a metric space with finite diameter. Let \(f\colon\mathbb N\to (0,\infty)\) be a non-increasing function with \(\sum_{n=0}^\infty f(n)<\infty\) such that, for any \(k\geq 1\), there are \(M,N>0\) so that \(Mf(r)\leq f(kr)\leq Nf(r)\) for all \(r\). Define a new length metric on the Cayley graph by giving an edge \(e\) length \(f(n)\) if the distance from \(e\) to the identity element is \(n\). This new metric space \(\Gamma_f\) has finite diameter. Its completion \(\overline\Gamma_f\) is a Floyd completion and \(\partial_f G=\overline\Gamma_f\setminus\Gamma_f\) is a Floyd boundary. The left translation action of \(G\) on itself induces actions on \(\overline\Gamma_f\) and \(\partial_fG\) by biLipschitz maps. Floyd proved that if \(G\) is a geometrically finite Kleinian group and \(f(n)=\frac{1}{n^2+1}\), then there is a continuous equivariant map from the Floyd boundary to the limit set. The main result of the current paper generalizes this result to the case of relatively hyperbolic groups. Given a relatively hyperbolic group \(G\), there is some \(0<\lambda<1\), such that for the function \(f(n)=\lambda^n\), there is a continuous equivariant map from the Floyd boundary \(\partial_fG\) to the Bowditch boundary. This in particular implies the Floyd boundary is non-trivial. To facilitate the proof of the main theorem, the author proposes two new definitions of relatively hyperbolic groups, which are equivalent to the known ones. They serve as bridges between the geometric definition (actions on fine graphs) and the dynamics definition (convergence action). Another interesting result is the so called Attractor Sum Theorem. It says the following: given a convergence action of a locally compact group \(G\) on a compactum \(\Lambda\), and a proper cocompact action of \(G\) on a locally compact Hausdorff sp…
reviewed by: Xiangdong Xie
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 6125049
zbMATH Keywords: Floyd completions
zbMATH Keywords: relatively hyperbolic groups
zbMATH Keywords: convergence group actions
zbMATH Keywords: locally compact groups
zbMATH Keywords: Cayley graphs
zbMATH Keywords: geometrically finite Kleinian groups
MaRDI profile type: Publication
OpenAlex ID: W1993482570
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DOI: 10.1007/S00039-012-0175-6
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description: scientific article; zbMATH DE number 6125049