Geodesic intersections and isoxial Fuchsian groups

TL;DR

This paper investigates the relationship between geodesic intersections, isoxial Fuchsian groups, and their commensurability classes, providing evidence that the axes of hyperbolic elements can determine the class for most Fuchsian groups.

Contribution

The paper proves that the conjecture linking axes of hyperbolic elements to commensurability class holds for almost all Fuchsian groups, extending previous results to a broader class.

Findings

The axes of hyperbolic elements determine the commensurability class for most Fuchsian groups.

The method used explains why the conjecture fails for arithmetic groups.

The paper verifies the conjecture for groups of the second kind and almost all Fuchsian groups.

Abstract

The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified in for groups of the second kind by G. Mess and for arithemetic groups by by D. Long and A. Reid. Here we show that the conjecture holds for almost all Fuchsian groups and explain why our method fails for arithemetic groups.