The growth of a fixed conjugacy class in negative curvature

Abstract

Let $M$ be a compact closed manifold of variable negative curvature. Fix an element $\operatorname{id} \neq \gamma$ in the fundamental group $\Gamma$ of $M$, and denote the set of elements in $\Gamma$ that are conjugate to $\gamma$ by $\operatorname{Conj}_\gamma$. For two points $x, y$ in the universal cover of $M$, we obtain asymptotics for the number of $\operatorname{Conj}_\gamma$--orbits of $y$ that lie in a ball of radius $T$ centered at $x$, as $T$ tends to infinity. If $M$ is two-dimensional, or of dimension $n \geq 3$ and curvature bounded above by $-1$ and below by $-(\frac{n-1}{n-2})^2$, we find an exponentially small error term for this count.