Morse elements in Garside groups are strongly contracting

Calvez, Matthieu; Wiest, Bert

doi:10.2140/agt.2024.24.4545

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Morse elements in Garside groups are strongly contracting

Matthieu Calvez and Bert Wiest

Algebraic & Geometric Topology 24 (2024) 4545–4574

DOI: 10.2140/agt.2024.24.4545

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 $G$ .

Keywords

Garside groups, Morse property, strong contraction property, injective metric space

Mathematical Subject Classification

Primary: 20F36

Secondary: 20F65

References

Publication

Received: 18 August 2022

Revised: 5 April 2023

Accepted: 12 May 2023

Published: 17 December 2024

Authors

Matthieu Calvez
IRISIB Brussels Belgium

Bert Wiest
IRMAR - UMR CNRS 6625 Université de Rennes Rennes France

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Volume 24, issue 8 (2024)