Two generalisations of Leighton's Theorem

Abstract

Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first generalisation, which we refer to as the ball-restricted version, restricts how balls of a given size in the universal cover can map down to the two finite graphs when factoring through the common finite cover -this answers a question of Neumann from [11]. Secondly, we consider covers of graphs of spaces, and of more general objects.We also compute upper bounds for the sizes of the finite covers obtained in Leighton's Theorem and its generalisations. An appendix by Gardam and Woodhouse provides an alternative proof of the ball-restricted version, that uses Haar measure instead of groupoids.

Small non-Leighton two-complexes

“…Alternative proofs and various generalisations of this result can be found, e.g., in [Neu10], [BaK90], [SGW19], [Woo21], [BrS21], and references therein. Does a similar result hold for any CW-complexes, i.e.…”

Small non-Leighton two-complexes

“…Alternative proofs and generalisations of this result can be found, e.g., in [Neu10], [BaK90], [SGW19], [Woo21], [BrS21], and references therein. For two-dimensional CW-complexes (and cell coverings) the similar assertion is false:…”