Counting conjugacy classes in groups with contracting elements

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 scriptCfalse(o,nfalse)$\mathcal {C}(o, n)$ (respectively, C′(o,n)$\mathcal {C}^{\prime }(o, n)$) the set of (respectively, primitive) conjugacy classes of algebraic length at most n$n$ for a basepoint o$o$. The main result is the following asymptotic formula: ♯scriptC(o,n)≍♯C′(o,n)≍expfalse(ωfalse(Gfalse)nfalse)n.\begin{equation*}\hskip7pc \sharp \mathcal {C}(o, n) \asymp \sharp \mathcal {C}^{\prime }(o, n) \asymp \frac{\exp (\omega (G)n)}{n}.\hskip-7pc \end{equation*}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 a 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 element in mapping class groups has their Teichmüller axis contained in the principal stratum.