Orbital Counting in Conjugacy Classes

TL;DR

This paper develops asymptotic formulas for counting points in orbits of discrete groups acting on negatively curved spaces, focusing on elements within specific conjugacy classes.

Contribution

It introduces new asymptotic counting results for conjugacy class-restricted orbits in negatively curved and $CAT(-1)$ spaces, extending classical group action analysis.

Findings

Asymptotic formulas for orbit counts in conjugacy classes

Results apply to convex cocompact actions on $CAT(-1)$ spaces

Generalizes classical orbit counting to restricted conjugacy classes

Abstract

In this article we consider a restricted orbital counting problem for the action of certain discrete groups on suitable spaces. In particular, we present asymptotics for counting those points in an orbit restricted to a single conjugacy class. A classical example would be cocompact actions of a discrete group acting isometrically on a simply connected manifold with pinched negative curvature. More generally, we obtain results for convex cocompact actions on $CAT(-1)$ spaces.