# Transverse properties of parabolic subgroups of Garside groups

## TL;DR

This paper investigates the structure of parabolic subgroups in Garside groups, establishing minimal length transversals, regular language representations, and properties of coset growth and projections.

## Contribution

It introduces a minimal length transversal for parabolic subgroups in Garside groups and shows the coset growth series is rational, also analyzing projection properties.

## Key findings

- Existence of a minimal length transversal with a regular language structure.
- The coset growth series of parabolic subgroups is rational.
- Garside groups have fellow projections but not bounded projections on parabolic subgroups.

## Abstract

Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ 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 $\theta \in T$ is of minimal length in its right-coset, $H \theta$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. 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 $H$.