Leighton’s theorem : Extensions, limitations and quasitrees

Bridson, Martin R; Shepherd, Sam

doi:10.2140/agt.2022.22.881

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Leighton's theorem: Extensions, limitations and quasitrees

Martin R Bridson and Sam Shepherd

Algebraic & Geometric Topology 22 (2022) 881–917

DOI: 10.2140/agt.2022.22.881

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.

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

Volume 22, issue 2 (2022)