Transverse properties of parabolic subgroups of Garside groups

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