Transverse properties of parabolic subgroups of Garside groups

Abstract

Let G be a Garside group endowed with the generating set ${\cal S}$ of non-trivial simple elements, and let H be a parabolic subgroup of G. We determine a transversal T of H in G such that each θ ∈ T is of minimal length in its right-coset, Hθ, for the word length with respect to ${\cal S}$. We show that there exists a regular language L on ${\cal S} \cup {{\cal S}^{- 1}}$ and a bijection ev: L → T satisfying $\lg \left(U \right) = {\lg _{\cal S}}\left({ev\left(U \right)} \right)$ for all U ∈ L. From this we deduce that the coset growth series of H in G is rational. Finally, we show that G has fellow projections on H but does not have bounded projections on H.