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International audienceWe 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

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