Generalized word metrics and a Mazur–Ulam-type theorem for gyrogroups

Teerapong Suksumran; Teerapong Suksumran

doi:10.3934/era.2026181

Electronic Research Archive

2026, Volume 34, Issue 6: 4037-4050. doi: 10.3934/era.2026181

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Research article

Generalized word metrics and a Mazur–Ulam-type theorem for gyrogroups

Teerapong Suksumran ^,

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received: 26 February 2026 Revised: 14 April 2026 Accepted: 22 April 2026 Published: 14 May 2026
Primary 05C25; Secondary 20N05, 51F99

We investigate the metric geometry of gyrogroups, a class of group-like structures whose binary operation is generally nonassociative. In particular, we extend the notion of the word metric from finitely generated groups to gyrogroups. This extension enables any gyrogroup to be viewed as a metric space, providing a suitable framework for proving a Mazur–Ulam-type theorem and for analyzing its algebraic and combinatorial structure via the associated right Cayley graph in a manner analogous to the classical setting of groups.
- word metric,
- generating set,
- gyrogroup,
- Cayley graph,
- isometry
Citation: Teerapong Suksumran. Generalized word metrics and a Mazur–Ulam-type theorem for gyrogroups[J]. Electronic Research Archive, 2026, 34(6): 4037-4050. doi: 10.3934/era.2026181

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Abstract

We investigate the metric geometry of gyrogroups, a class of group-like structures whose binary operation is generally nonassociative. In particular, we extend the notion of the word metric from finitely generated groups to gyrogroups. This extension enables any gyrogroup to be viewed as a metric space, providing a suitable framework for proving a Mazur–Ulam-type theorem and for analyzing its algebraic and combinatorial structure via the associated right Cayley graph in a manner analogous to the classical setting of groups.

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