Garside shadows and biautomatic structures in Coxeter groups

TL;DR

The paper extends the Osajda–Przytycki biautomaticity framework for Coxeter groups by introducing voracious projections and voracious languages relative to any finite Garside shadow $B$ in a Coxeter system $(W,S)$. It proves that $(S,\\mathcal{V}_B)$ forms a biautomatic structure for $W$ when $B$ is finite, and shows that the original Osajda–Przytycki construction is recovered in the special case $B = L$ (the low elements). The work also establishes regularity of $\\mathcal{V}_B$ for finite $B$ via a finite automaton and proves the fellow-traveller properties using a generalized parallel-wall framework. Collectively, these results answer a question of Hohlweg and Parkinson and broaden the toolkit for algorithmic analysis of Coxeter groups through Garside-shadow-based automata and languages.

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 $g \in W$, 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.

Figures (4)

• Figure 1: Proof of Lemma \ref{['v-ineq-lemma']}.
• Figure 2: Proof that $v$ is accepted by $\mathcal{A}_B$.
• Figure 3: Proof that $v$ belongs to $\mathcal{V}_B$.
• Figure 4: Proof of Proposition \ref{['propFTPb']}.

Theorems & Definitions (25)

• Theorem 1.1
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Definition 3.4
• Proposition 3.5
• Definition 4.1
• Definition 4.2
• Proposition 4.3