The growth of a fixed conjugacy class in negative curvature

TL;DR

This paper studies the asymptotic growth of conjugacy classes in the fundamental group of negatively curved manifolds, providing precise counts with exponential error terms in certain dimensions and curvature bounds.

Contribution

It derives asymptotic formulas for the growth of conjugacy class orbits in negatively curved manifolds, including explicit error estimates in specific geometric settings.

Findings

Asymptotic counts for conjugacy class orbits in negatively curved manifolds.

Exponential error terms for the count in 2D and certain higher dimensions.

Results applicable to manifolds with curvature bounds between -1 and a specified negative constant.

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.