Uniform growth in small cancellation groups

Abstract An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. There are two consequences: firstly, there is a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. Secondly, the uniform uniform exponential growth rate of a classical ‐small cancellation group, for sufficiently small , is bounded from below by a universal positive constant. We give a similar result for uniform entropy‐cardinality estimates. This yields an explicit upper bound on the number of isomorphism classes of marked ‐hyperbolic ‐small cancellation groups of uniformly bounded entropy in terms of and the entropy bound.