Statistics in conjugacy classes in free groups

TL;DR

This paper investigates the statistical properties of conjugacy classes in free groups acting on CAT(-1) spaces, revealing a modified variance in the central limit theorem when restricting to a conjugacy class.

Contribution

It introduces a new statistical framework for conjugacy classes in free groups acting on negatively curved spaces, including a central limit theorem with an adjusted variance.

Findings

Established a central limit theorem for conjugacy classes in free groups.

Discovered the variance doubles when restricting to a conjugacy class.

Provided new insights into the statistical behavior of free group actions.

Abstract

In this paper, we establish statistical results for a convex co-compact action of a free group on a CAT($-1$) space where we restrict to a non-trivial conjugacy class in the group. In particular, we obtain a central limit theorem where the variance is twice the variance that appears when we do not make this restriction.