# On the displacement of generators of free Fuchsian groups

## TL;DR

This paper establishes a new inequality for generator displacements in free Fuchsian groups, leading to improved geometric bounds on hyperbolic surfaces, including the Margulis constant and closed curve lengths.

## Contribution

It introduces a two-dimensional analogue of the $ ext{log}(2k-1)$ Theorem for Kleinian groups, providing new quantitative geometric results.

## Key findings

- Derived a displacement inequality for free Fuchsian group generators
- Improved bounds on the Margulis constant for hyperbolic surfaces
- Enhanced estimates on lengths of closed geodesics

## Abstract

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the $\log (2k-1)$ Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of closed curves, which improves a result of Buser's.