Morse elements in Garside groups are strongly contracting

TL;DR

This paper demonstrates that in Garside groups with cyclic center, the axes of Morse elements are strongly contracting and act loxodromically, extending known results from braid groups to a broader class of Garside groups.

Contribution

It generalizes the strong contraction property of Morse elements from braid groups to all Garside groups of finite type with cyclic center.

Findings

Axes of Morse elements are strongly contracting in Garside groups.

Morse elements act loxodromically on the additional length graph.

The results apply to braid groups modulo their center and more general Garside groups.

Abstract

We prove that in the Cayley graph of any braid group modulo its center $B_n/Z(B_n)$, equipped with Garside's generating set, the axes of all pseudo-Anosov braids are strongly contracting. More generally, we consider a Garside group $G$ of finite type with cyclic center. We prove that in the Cayley graph of $G/Z(G)$, equipped with the Garside generators, the axis of any Morse element is strongly contracting. As a consequence, we prove that Morse elements act loxodromically on the additional length graph of $G$.