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Gaussian groups and Garside groups, two generalisations of Artin groups. (English) Zbl 1030.20021
It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.
Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lower common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lcm’s satisfy an additional symmetry condition. A Gaussian group and a Garside group are respectively the group of fractions of a Gaussian monoid and of a Garside monoid. Braid groups and, more generally, finite Coxeter type Artin groups are Garside groups. We determine algorithmic criteria in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that these algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter type Artin group that can be explicitly determined.

MSC:
20F05 Generators, relations, and presentations of groups
20F36 Braid groups; Artin groups
20M05 Free semigroups, generators and relations, word problems
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F65 Geometric group theory
20-04 Software, source code, etc. for problems pertaining to group theory
20M35 Semigroups in automata theory, linguistics, etc.
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