Journal of Modern Dynamics (JMD)

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

Pages: 269 - 290, Issue 2, June 2013      doi:10.3934/jmd.2013.7.269

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Stéphane Sabourau - Université Paris-Est, Laboratoire d’Analyse et Mathématiques Appliquées (UMR 8050), UPEC, UPEMLV, CNRS, F-94010, Créteil, France (email)

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.

Keywords:  Exponential growth rate, entropy, critical exponent, asymptotic group theory, Gromov-hyperbolic spaces, normal subgroups, covers, growth-tightness.
Mathematics Subject Classification:  Primary: 20F69; Secondary: 20F67, 20E07, 53C23.

Received: December 2012;      Available Online: September 2013.