The Language of Geodesics for Garside Groups
نویسندگان
چکیده
We prove that the language of all geodesics of any Garside group, with respect to the generating set of divisors of the Garside element, forms a regular language. In particular, the braid groups admit generating sets where the associated language of geodesics is regular.
منابع مشابه
Gaussian groups and Garside groups, two generalisations of Artin groups
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...
متن کاملGrowth of Minimal Word-length in Garside Groups
The Garside group, as a generalization of braid groups and Artin groups of finite types, is defined as the group of fractions of a Garside monoid. We show that the semidirect product of Garside monoids is a Garside monoid. We use the semidirect product Z ⋉ G of the infinite cyclic group Z and the cartesian product G of a Garside group G to study the properties of roots and powers of elements in...
متن کاملParabolic subgroups of Garside groups
A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...
متن کاملCone Types and Geodesic Languages for Lamplighter Groups and Thompson’s Group F Sean Cleary, Murray Elder, and Jennifer Taback
We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...
متن کاملCone types and geodesic languages for lamplighter groups and Thompson’s group F
We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...
متن کامل