A conjecture about Artin-Tits groups

TL;DR

This paper conjectures a simplified solution to the word problem in Artin-Tits groups, relates it to hyperbolicity, proves it for type FC, and explores potential extensions and approaches.

Contribution

It introduces a new conjecture about solving the word problem in Artin-Tits groups without trivial factors and proves it for type FC groups.

Findings

Proves the conjecture for Artin-Tits groups of type FC.

Proposes a syntactic approach for right-angled cases.

Links the conjecture to a form of hyperbolicity.

Abstract

We conjecture that the word problem of Artin-Tits groups can be solved without introducing trivial factors ss^{-1} or s^{-1}s. Here we make this statement precise and explain how it can be seen as a weak form of hyperbolicity. We prove the conjecture in the case of Artin-Tits groups of type FC, and we discuss various possible approaches for further extensions, in particular a syntactic argument that works at least in the right-angled case.