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A Garside presentation for Artin-Tits groups of type \(\widetilde C_n\). (English. French summary) Zbl 1260.20056
Summary: We prove that an Artin-Tits group of type \(\widetilde C\) is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type \(\widetilde C\), and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type \(\widetilde A\) under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20M05 Free semigroups, generators and relations, word problems
20F55 Reflection and Coxeter groups (group-theoretic aspects)
57M07 Topological methods in group theory
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