Orbit counting in conjugacy classes for free groups acting on trees

TL;DR

This paper investigates the asymptotic behavior of conjugacy class elements in the fundamental group of a finite metric graph acting on its universal cover, providing precise growth rates and error estimates under certain conditions.

Contribution

It introduces new asymptotic formulas for counting conjugacy class elements with bounded displacement in the universal covering tree of a finite metric graph, including polynomial error terms.

Findings

Asymptotic formulas for conjugacy class element counts

Polynomial error term under additional assumptions

Conditions on graph and edge lengths for results

Abstract

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most $T$. Under a mild extra assumption we also obtain a polynomial error term.