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 of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system (“reduction”) that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin–Tits monoids, and provide partial results and numerical evidence supporting such conjectures.