Counting conjugacy classes in groups with contracting elements

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