On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts

TL;DR

This paper explores the limitations of existing proofs of Leighton's Graph Covering Theorem, introduces a new example of non-constructible common covers, and proves that normal virtual retracts are virtual direct summands, with applications to various group properties.

Contribution

It provides a novel example of finite covers not obtainable via tree lattice techniques and establishes a general criterion for when a commensuration can be induced by a co-commensuration, also proving a conjecture about virtual retracts.

Findings

An explicit example of a common finite cover not derived from tree lattice methods.

A necessary and sufficient condition for inducing a commensuration via a co-commensuration.

Proof that normal virtual retracts are virtual direct summands.

Abstract

Leighton's Graph Covering Theorem states that if two finite graphs have the same universal covering tree, then they also have a common finite degree cover. Bass and Kulkarni gave an alternative proof of this fact using tree lattices. We give an example of two graphs that admit a common finite cover which can not be obtained using tree lattice techniques. If two groups embed as finite index subgroups, we say they are co-commensurable. Our example comes from an explicit commensuration that cannot be induced by a co-commensuration. Next we state and prove a general theorem that gives necessary and sufficient conditions for when a commensuration can be induced by a co-commensuration. The developed machinery is then used to show that normal virtual retracts are virtual direct summands, answering a question of Merladet and Minasyan. In an appendix, applications to commensurating graphs of groups, biautomaticity, and hereditary conjugacy separability are given.