Fundamental domains in ${\rm PSL}(2,{\mathbb R})$ for Fuchsian groups

TL;DR

This paper establishes a precise criterion for identifying fundamental domains in ${ m PSL}(2,{ m R})$ and $T^1{ m H}^2$ associated with Fuchsian groups, linking algebraic and geometric perspectives.

Contribution

It provides a necessary and sufficient condition for a set to serve as a fundamental domain for Fuchsian groups in both algebraic and geometric contexts.

Findings

Derived a criterion linking fundamental domains in ${ m PSL}(2,{ m R})$ and ${ m H}^2$

Unified algebraic and geometric descriptions of Fuchsian group actions

Enhanced understanding of fundamental domain structures in hyperbolic geometry

Abstract

In this paper, we provide a necessary and sufficient condition for a set in ${\rm PSL}(2,{\mathbb R})$ or in $T^1{\mathbb H}^2$ to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane ${\mathbb H}^2$.