Growth of quotients ofgroups acting by isometries on Gromov-hyperbolic spaces

Stéphane Sabourau

doi:10.3934/jmd.2013.7.269

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2013, Volume 7, Issue 2: 269-290. Doi: 10.3934/jmd.2013.7.269

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Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces

Stéphane Sabourau^1,

1.
Université Paris-Est, Laboratoire d’Analyse et Mathématiques Appliquées (UMR 8050), UPEC, UPEMLV, CNRS, F-94010, Créteil

Received: November 30, 2012

Published: August 2013

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Abstract

We show that every group $G$ with no cyclic subgroup of finite index that acts properly and cocompactly by isometries on a proper geodesic Gromov-hyperbolic space $X$ is growth-tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result unifies and extends previous works of Arzhantseva-Lysenok and Sambusetti using a geometric approach.

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Mathematics Subject Classification: Primary: 20F69; Secondary: 20F67, 20E07, 53C23.

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