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Leighton's theorem: Extensions, limitations and quasitrees

Martin R Bridson and Sam Shepherd

Algebraic & Geometric Topology 22 (2022) 881–917
Abstract

Leighton’s theorem states that if there is a tree T that covers two finite graphs G1 and G2, then there is a finite graph Ĝ that is covered by T and covers both G1 and G2. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to nonregular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.

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Keywords
quasitrees, covering spaces, Leighton's theorem
Mathematical Subject Classification
Primary: 05C25, 20F65, 20F67
References
Publication
Received: 9 September 2020
Revised: 13 January 2021
Accepted: 1 February 2021
Published: 3 August 2022
Authors
Martin R Bridson
Mathematical Institute
University of Oxford
Oxford
United Kingdom
Sam Shepherd
Department of Mathematics
Vanderbilt University
Nashville, TN
United States