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arXiv · 1810.02969

Counting conjugacy classes in groups with contracting elements

Ilya Gekhtman·Wen-yuan Yang
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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: $$\sharp \mathcal C(o, n) \asymp \sharp \mathcal C'(o, n) \asymp \frac{\exp(ω(G)n)}{n}.$$ 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.

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Ilya Gekhtman, Wen-yuan Yang. 2022-02-16. Counting conjugacy classes in groups with contracting elements. https://arxiv.org/abs/1810.02969

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