#### Volume 22, issue 2 (2022)

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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 ${G}_{1}$ and ${G}_{2}$, then there is a finite graph $Ĝ$ that is covered by $T$ and covers both ${G}_{1}$ and ${G}_{2}$. 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
##### Publication
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