Garside shadows and biautomatic structures in Coxeter groups

2025/05/27 by Santos, Fabricio Dos · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Group Theory (math.GR) #Random Matrices and Applications #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2505.21718

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 2022, Osajda and Przytycki showed that any Coxeter group W is biautomatic. Key to their proof is the notion of voracious projection of an element g ∈ W, which is used iteratively to construct a biautomatic structure for W: the voracious language. In this article, we generalize these two notions by defining them for any Garside shadow B in a Coxeter system (W,S). This leads to the result that any finite Garside shadow in (W,S) can be used to construct a biautomatic structure for W. In addition, we show that for the Garside shadow L of low elements, the biautomatic structure obtained corresponds to the original voracious language of Osajda and Przytycki. These results answer a question of Hohlweg and Parkinson.

Citations

• A Pair of Garside Shadows
• Coxeter groups are biautomatic
• Cone Types, Automata, and Regular Partitions in Coxeter Groups
• Characterizing Biautomatic Groups
• Small roots, low elements, and the weak order in Coxeter groups
• Garside families in Artin-Tits monoids and low elements in Coxeter groups
• Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Cited by

• Automaticity of non-positively curved k-fold triangle groups

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