Two generalisations of Leighton’s theorem (with an appendix by Giles Gardam and Daniel J. Woodhouse)

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 symmetry-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 (2010). Secondly, we consider covers of graphs of spaces (or of more general objects), which leads to an even more general version of Leighton’s theorem.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 symmetry-restricted version, that uses Haar measure instead of groupoids.