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 $C^{\prime \prime }(\hat{I}»)$-small cancellation group, for sufficiently small $\hat{I}»$, 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 isomorphism class of marked $\hat{I}´$-hyperbolic $C^{\prime \prime }(\hat{I}»)$-small cancellation groups of uniformly bounded entropy in terms of $\hat{I}´$ and the entropy bound.

47 pages, author accepted manuscript