Uniform growth in small cancellation groups

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''(λ)-small cancellation group, for sufficiently small λ, 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 δ-hyperbolic C''(λ)-small cancellation groups of uniformly bounded entropy in terms of δ and the entropy bound.