Uniform growth in small cancellation groups

TL;DR

This paper investigates uniform exponential growth in acylindrically hyperbolic groups, showing stability under small cancellation quotients and providing explicit bounds related to hyperbolic small cancellation groups.

Contribution

It proves that groups with uniform exponential growth acting acylindrically are closed under certain small cancellation quotients and establishes explicit bounds for hyperbolic small cancellation groups.

Findings

Existence of a finitely generated acylindrically hyperbolic group with uniform exponential growth and arbitrarily large torsion balls.

Lower bound on the growth rate for classical small cancellation groups with small parameters.

Explicit bounds on the isomorphism classes of hyperbolic small cancellation groups with bounded entropy.

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 $C''(\lambda)$-small cancellation group, for sufficiently small $\lambda$, 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 $\delta$-hyperbolic $C''(\lambda)$-small cancellation groups of uniformly bounded entropy in terms of $\delta$ and the entropy bound.