Length functions on groups and actions on graphs
Length functions on groups and actions on graphs
We study generalizations of Chiswell’s theorem that 0-hyperbolic Lyndon length functions on groups always arise as based length functions of the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces 0-hyperbolicity with δ-hyperbolicity. We then propose a set of axioms for the length function on a finitely generated group that ensures the function is bi-Lipschitz equivalent to a (or any) length function of the group acting on its Cayley graph with respect to some finite generating set.
20E08, 20F65, 20F67, math.GR, Lyndon length functions, Bi-Lipschitz equivalence, group actions on graphs, hyperbolicity, Chiswell’s theorem
2269-2281
Collins, Matthew
39c341a9-3dab-40e3-9d5a-b77cdb360ffd
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
2024
Collins, Matthew
39c341a9-3dab-40e3-9d5a-b77cdb360ffd
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Collins, Matthew and Martino, Armando
(2024)
Length functions on groups and actions on graphs.
Communications in Algebra, 52 (6), .
(doi:10.48550/arXiv.2307.10760).
Abstract
We study generalizations of Chiswell’s theorem that 0-hyperbolic Lyndon length functions on groups always arise as based length functions of the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces 0-hyperbolicity with δ-hyperbolicity. We then propose a set of axioms for the length function on a finitely generated group that ensures the function is bi-Lipschitz equivalent to a (or any) length function of the group acting on its Cayley graph with respect to some finite generating set.
Text
2307.10760v1
- Author's Original
Available under License Other.
Text
Length functions on groups and actions on graphs
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Accepted/In Press date: 5 December 2023
e-pub ahead of print date: 28 December 2023
Published date: 2024
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Publisher Copyright:
© 2023 The Author(s). Published with license by Taylor & Francis Group, LLC.
Keywords:
20E08, 20F65, 20F67, math.GR, Lyndon length functions, Bi-Lipschitz equivalence, group actions on graphs, hyperbolicity, Chiswell’s theorem
Identifiers
Local EPrints ID: 485036
URI: http://eprints.soton.ac.uk/id/eprint/485036
ISSN: 0092-7872
PURE UUID: fc6db7cf-5659-411e-8d48-b70358aa6655
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Date deposited: 28 Nov 2023 17:50
Last modified: 06 Jun 2024 01:46
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Author:
Matthew Collins
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