Rewriting systems in sufficiently large Artin-Tits groups

TL;DR

This paper proves Dehornoy's conjecture for sufficiently large Artin-Tits groups, showing that words representing the identity can be simplified using specific rewriting rules without trivial factors.

Contribution

It extends the validity of Dehornoy's conjecture to a new class of Artin-Tits groups, specifically those of sufficiently large type.

Findings

Conjecture verified for sufficiently large Artin-Tits groups

Rewriting systems can simplify words without trivial factors

Supports broader applicability of Dehornoy's conjecture

Abstract

A conjecture of Dehornoy claims that, given a presentation of an Artin-Tits group, every word that represents the identity can be transformed into the trivial word using the braid relations, together with certain rules (between pairs of words that are not both positive) that can be derived directly from the braid relations, as well as free reduction, but without introducing trivial factors $ss^{-1} $ or $s^{-1} s$. This conjecture is known to be true for Artin-Tits groups of spherical type or of FC type. We prove the conjecture for Artin--Tits groups of sufficiently large type.