Garside shadows and biautomatic structures in Coxeter groups

TL;DR

This paper generalizes the concept of voracious projection and biautomatic structures in Coxeter groups using Garside shadows, showing that finite Garside shadows can produce biautomatic structures and connecting to previous work.

Contribution

It introduces Garside shadows into the construction of biautomatic structures in Coxeter groups, extending prior methods and answering open questions.

Findings

Finite Garside shadows yield biautomatic structures in Coxeter groups

Garside shadow of low elements recovers original voracious language

Results generalize and unify previous constructions in the literature

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.