The language of geodesics for Garside groups
The paper studies a certain class of finitely generated groups from the point of view of regular language. If a finitely generated group fixes a generating set, then the group admits a metric structure called word metric. There is a basic question raised by J. Cannon, under which conditions can the set of all geodesics be a regular language with respect to a generating set, which implies that it can be accepted by a finite state automaton. This paper studies the Garside groups, which is a class of finitely generated groups. Roughly speaking, they are characterised by some properties similar to the braid groups. In particular, the braid groups and also the Artin groups of finite type are Garside groups. The main theorem of this paper is that the language of geodesics of a Garside group with respect to the generating set of simple divisors of \(\Delta\) is regular, where \(\Delta\) is a similar element to the half twist in the braid groups. Regularity of language is shown by constructing an explicit automaton. Regularity is reflected by the rationality of some growth functions including spherical growth series. Let us define the geodesic growth series by: \[ l(z)=\sum_{z\in G} p(g)z^{|g|} \] where \(p(g) \in {\mathbb N}\) is the number of geodesics whose end points correspond to \(g\). The authors also show that if the language is regular, then \(l(z)\) is rational. In particular it follows that the geodesic growth series of a Garside group with respect to the generating set of simple divisors of \(\Delta\) is rational.
scientific article; zbMATH DE number 2131991
scientific article