Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces
Stéphane Sabourau
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 Gromov-hyperbolic spaces asymptotic group theory critical exponent normal subgroups covers growth-tightness.

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