Morse elements in Garside groups are strongly contracting

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 $G$.

25 pages, 9 figures

References in corpus (4)

• On strongly quasiconvex subgroups
• Geodesics in the mapping class group
• Morse subsets of CAT(0) spaces are strongly contracting
• Divergence of finitely presented subgroups of CAT(0) groups

Cited by in corpus (1)

• Random walks and contracting elements I: Deviation inequality and Limit laws