Coverings of Profinite Graphs

TL;DR

This paper introduces a theory of coverings for profinite graphs, defining universal covers and fundamental groups, and establishing parallels with classical graph covering theory.

Contribution

It develops a new covering theory for profinite graphs, including the existence and uniqueness of universal covers and the definition of profinite fundamental groups.

Findings

Universal cover of a profinite graph always exists and is unique.

Connected covers are universal if and only if their fundamental group is trivial.

Profinite fundamental groups are well-defined and analogous to classical cases.

Abstract

We define a covering of a profinite graph to be a projective limit of a system of covering maps of finite graphs. With this notion of covering, we develop a covering theory for profinite graphs which is in many ways analogous to the classical theory of coverings of abstract graphs. For example, it makes sense to talk about the universal cover of a profinite graph and we show that it always exists and is unique. We define the profinite fundamental group of a profinite graph and show that a connected cover of a connected profinite graph is the universal cover if and only if its profinite fundamental group is trivial.