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 $\mathcal{C}(o,n)$ (resp. ${\mathcal{C}}^{\prime }(o,n)$) 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: $$♯\mathcal{C}(o,n)\asymp ♯{\mathcal{C}}^{\prime }(o,n)\asymp \frac{\exp (\omega (G)n)}{n}.$$ 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 Teichmüller axis contained in the principal stratum.

arXiv commentVersion 3: 41 pages, 5 figures; Version accepted to Journal of Topology, many improvements and clarification