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Revisiting Leighton’s theorem with the Haar measure

Part of: Low-dimensional topology Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  13 January 2020

DANIEL J. WOODHOUSE
DANIEL J. WOODHOUSE*
Affiliation:
Postal address: Mathematics Department Technion – Israel Institute of Technology HAIFA - 32000, ISRAEL. e-mail: [email protected]
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Abstract

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

MSC classification

Primary: 57M10: Covering spaces
Secondary: 20E08: Groups acting on trees 20E05: Free nonabelian groups 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 615 - 623
Copyright
© Cambridge Philosophical Society 2020

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Footnotes

Present address: Mathematical Institute University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford, OX2 6GG.

†

Supported by the Israel Science Foundation (grant 1026/15).

References

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