Garside shadows and biautomatic structures in Coxeter groups

Fabricio Dos Santos
The University of Sydney

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 talk, I will explain how to 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 particular, 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.