Two generalisations of Leighton's theorem (with an appendix by Giles Gardam and Daniel J. Woodhouse)
type: Item
instance of: scholarly article
title: Two generalisations of Leighton's theorem (with an appendix by Giles Gardam and Daniel J. Woodhouse)
DOI: 10.4171/GGD/682
author: Sam Shepherd
published in: Groups, Geometry, and Dynamics
publication date: 2022-11-29T00:00:00Z
review text: Summary: 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.
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 7625249
zbMATH Keywords: Leighton's theorem
zbMATH Keywords: covering spaces
Wikidata QID: Q114986255
MaRDI profile type: Publication
cites work: Finite common coverings of pairs of regular graphs
cites work: Simple groups of automorphisms of trees determined by their actions on finite subtrees
cites work: Uniform Tree Lattices
cites work: Quasi-isometric classification of non-geometric 3-manifold groups
cites work: Q4720067
cites work: Finite common coverings of graphs
cites work: Generalized Baumslag-Solitar groups: rank and finite index subgroups
cites work: Lattices in rank one Lie groups over local fields
cites work: Majoration explicite de l'ordre maximum d'un élément du groupe symétrique
cites work: On Leighton's graph covering theorem
cites work: Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups
cites work: Revisiting Leighton’s theorem with the Haar measure
Recommended article: Leighton's theorem : extensions, limitations and quasitrees
Recommended article: Q3370142
Recommended article: Q5201328
Recommended article: Covering theory for graphs of groups
Recommended article: On Leighton's graph covering theorem
Recommended article: Q6077701
Recommended article: Q3198545
Recommended article: Q4234656
Recommended article: Semi-edges, reflections and Coxeter groups
Recommended article: Covering group theory for topological groups
description: scientific article; zbMATH DE number 7625249