Multifraction reduction II: Conjectures for Artin–Tits groups

Abstract

Multifraction reduction is a new approach to the Word Problem for Artin-Tits groups and, more generally, for the enveloping group U (M) of a monoid M in which any two elements admit a greatest common divisor. This approach is based on a rewrite system R(M) that extends free group reduction. In this paper, we show that for R(M) to satisfy a weak form of convergence called semi-convergence is sufficient for solving the Word Problem for U (M), and we connect semi-convergence with other conditions involving R(M). We conjecture that all these properties are valid when M is an Artin-Tits monoid, and report about numerical experiments supporting this conjecture.