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Length functions on groups and actions on graphs

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
0092-7872
2269-2281
Collins, Matthew
39c341a9-3dab-40e3-9d5a-b77cdb360ffd
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
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), 2269-2281. (doi:10.48550/arXiv.2307.10760).

Record type: Article

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.

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More information

Accepted/In Press date: 5 December 2023
e-pub ahead of print date: 28 December 2023
Published date: 2024
Additional Information: 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
ORCID for Armando Martino: ORCID iD orcid.org/0000-0002-5350-3029

Catalogue record

Date deposited: 28 Nov 2023 17:50
Last modified: 06 Jun 2024 01:46

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Contributors

Author: Matthew Collins
Author: Armando Martino ORCID iD

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