2019/08/02 by Sam O. Shepherd, Shepherd, Sam, G. E. Gardam +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #05C70 (Secondary) #20E08 (Primary) 57M10 #20L05 #Advanced Operator Algebra Research #Connective tissue disorders research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1908.00830
openalex publication_date 2019/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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.