Journal of Combinatorial Algebra

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Volume 1, Issue 3, 2017, pp. 229–287
DOI: 10.4171/JCA/1-3-1

Published online: 2017-07-25

Multifraction reduction II: Conjectures for Artin–Tits groups

Patrick Dehornoy[1]

(1) Université de Caen and and Institut Universitaire de France, France

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.

Keywords: Artin–Tits monoid, Artin–Tits group, gcd-monoid, enveloping group, word problem, multifraction, reduction, semi-convergence, cross-confluence, tame reduction, van Kampen diagram, embeddability

Dehornoy Patrick: Multifraction reduction II: Conjectures for Artin–Tits groups. J. Comb. Algebra 1 (2017), 229-287. doi: 10.4171/JCA/1-3-1