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Thin groups of fractions. (English) Zbl 1058.20045
Cleary, Sean (ed.) et al., Combinatorial and geometric group theory. Proceedings of the AMS special session on combinatorial group theory, New York, NY, USA, November 4–5, 2000 and the AMS special session on computational group theory, Hoboken, NJ, USA, April 28–29, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2822-3/ pbk). Contemp. Math. 296, 95-128 (2002).
Summary: A number of properties of spherical Artin-Tits groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied [the author, Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 2, 267-306 (2002; Zbl 1017.20031)]. Here we investigate a wider class of groups of fractions, called ‘thin’, which are those associated with monoids where minimal common multiples exist, but they are not necessarily unique. Also, we allow units in the involved monoids. The main results are that all thin groups of fractions satisfy a quadratic isoperimetric inequality, and that, under some additional hypotheses, they admit an automatic structure.
For the entire collection see [Zbl 0990.00044].

MSC:
20M05 Free semigroups, generators and relations, word problems
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
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