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.

令G为赋予非平凡简单元素的生成集\（{\ cal S} \）的Garside群，令H为G的抛物线子群。我们确定的横向Ť的ħ在ģ使得每个θ∈Ť处于其右陪集，最小长度的Hθ，对于单词长度相对于\（{\ CAL S} \） 。我们证明存在\（{\ cal S} \ cup {{\ cal S} ^ {-1}} \）上的正则语言L和双射ev：L→T满足\（\ lg \ left（U \ right）= {\ lg _ {\ cal S}} \ left（{ev \ left（U \ right）} \ right）\）对于所有U∈L。由此我们可以推断出H在G中的陪伴生长系列是合理的。最后，我们表明，摹有研究员预测对^ h，但不会对有界预测^ h。