Revisiting Leighton's Theorem with the Haar Measure

TL;DR

This paper presents a new proof of Leighton's theorem using Haar measure, extends it to graphs with fins, and explores applications in pattern rigidity and quasi-isometric rigidity for free groups.

Contribution

It introduces a generalized proof of Leighton's theorem, including graphs with fins, and applies it to rigidity problems in geometric group theory.

Findings

New proof of Leighton's theorem using Haar measure

Extension of the theorem to graphs with fins

Quasi-isometric rigidity results for cyclic doubles of free groups

Abstract

Leighton's graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton's theorem that allows generalizations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen-Macura and Hagen-Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.