Fundamental groups as limits of discrete fundamental groups

TL;DR

This paper explores how the fundamental group of a metric space can be represented as an inverse limit of discrete fundamental groups, identifying conditions for isomorphism and providing counterexamples to weaker hypotheses.

Contribution

It establishes conditions under which the fundamental group equals an inverse limit of discrete groups and presents counterexamples to potential relaxations.

Findings

Certain mild conditions guarantee the isomorphism.

Counterexamples show the necessity of these conditions.

Clarifies the relationship between fundamental groups and their discrete approximations.

Abstract

In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypotheses.