# Parabolic subgroups acting on the additional length graph

## TL;DR

This paper studies the action of certain elements on the additional length graph of irreducible Artin--Tits groups, revealing properties about their elliptic actions, and uses these to analyze growth rates of braid groups.

## Contribution

It demonstrates that periodic and parabolic-preserving elements act elliptically on the additional length graph and constructs specific elements with bounded length to analyze growth rates.

## Key findings

- Periodic and parabolic-preserving elements act elliptically on the graph.
- Constructed elements with bounded length that generate free products with parabolic subgroups.
- The exponential growth rates of braid groups with respect to Garside generators tend to infinity.

## Abstract

Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the Garside generators, independently of $A$. This allows us to show that, in contrast with the Artin generators case, the sequence $\{\omega(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity.