What is a fundamental domain?

TL;DR

This paper examines various definitions of fundamental domains for Fuchsian groups, highlighting the need for an additional condition to distinguish true fundamental domains from mere fundamental sets.

Contribution

It clarifies the criteria that should be included in definitions of fundamental domains to ensure they are well-defined and distinct from fundamental sets.

Findings

Many existing definitions lack an extra condition.

Adding a condition ensures the domain differs from a fundamental set only by a null set.

The paper provides examples illustrating these differences.

Abstract

We show by means of various examples that many of the current definitions of the notion of fundamental domain of a Fuchsian group lack an extra condition ensuring that the domain differs from a measurable fundamental set at most by a null set.