Counting conjugacy classes in groups with contracting elements

TL;DR

This paper establishes an asymptotic formula for counting conjugacy classes in groups with contracting elements, revealing their growth rate and implications for group properties and generic element behavior.

Contribution

It provides the first asymptotic formula for conjugacy class counts in groups with contracting elements, extending understanding of their growth and properties.

Findings

Conjugacy classes grow asymptotically as exp(ω(G)n)/n.

The conjugacy growth series is transcendental for certain groups.

Exponential genericity of elements with specific geometric properties.

Abstract

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set of (resp. primitive) conjugacy classes of pointed length at most $n$ for a basepoint $o$. The main result is an asymptotic formula as follows: $$\sharp \mathcal C(o, n) \asymp \sharp \mathcal C'(o, n) \asymp \frac{\exp(\omega(G)n)}{n}.$$ A similar formula holds for conjugacy classes using stable length. As a consequence of the formulae, the conjugacy growth series is transcendental for all non-elementary relatively hyperbolic groups, graphical small cancellation groups with finite components. As by-product of the proof, we establish several useful properties for an exponentially generic set of elements. In particular, it yields a positive answer to a question of J. Maher that an exponentially generic elements in mapping class groups have their Teichm\"{u}ller axis contained in the principal stratum.