Morse elements in Garside groups are strongly contracting (Q6660981)

Let \(G\) be a finitely generated group, \(S\) a finite set of generators of \(G\) and let \(\Gamma(G)=\Gamma_{S}(G)\) be the Cayley graph of \(G\) associated to \(S\). If \(g\in G\) is an element of infinite order the axis \(\mathsf{axis}(g)\) is the set \(\langle g \rangle\) seen as a set of vertices of \(\Gamma(G)\).\N\NAn element \(g\) is Morse if this axis is quasi-isometrically embedded in \(\Gamma(G)\) and if for each pair of constants \((K,L)\), there exists a constant \(M_{g}^{(K,L)}\) such that every \((K,L)\)-quasigeodesic between two points of the \(\mathsf{axis}(g)\) travels in an \(M_{g}^{(K,L)}\)-neighborhood of the axis. A Garside group of finite type \(G\) is generated by a finite lattice \(\mathcal{D}\) with a top element called \(\Delta\). In a Garside group every element \(g \in G\) is represented by a unique word in a certain normal form, with letters in \(\mathcal{D}^{\pm 1}\); these normal form words represent geodesics in the Cayley graph of \(G\) with respect to \(\mathcal{D}\). In particular the center of \(G\) is infinite cyclic and is generated by some power \(\Delta^{e}\). If \(\overline{\Gamma}=\Gamma(G/Z(G), \mathcal{D})\), then an element \(g \in G\) of a Garside group \(G\) is called Morse if its axis in \(\overline{\Gamma}\) is Morse. The authors define \(\mathcal{X}\) to be the quotient of the Cayley graph \(\Gamma(G,\mathcal{D})\) under the right \(\langle \Delta \rangle\)-action (the graph \(\mathcal{X}\) is the 1-skeleton of the simplicial complex previously considered in [\textit{M. Bestvina}, Geom. Topol. 3, 269--302 (1999; Zbl 0998.20034); \textit{R. Charney} et al. Geom. Dedicata 105, 171--188 (2004; Zbl 1064.20044)]).\N\NThe main result in the paper under review is Theorem 1.1: Suppose \(G\) is a \(\Delta\)-pure Garside group of finite type. Suppose \(g\) is an element of \(G\) whose axis in \(\mathcal{X}\) (or, equivalently, in \(\overline{\Gamma}\)) is Morse. Then this axis is strongly contracting, both as a subset of \(\mathcal{X}\) and as a subset of \(\overline{\Gamma}\).\N\NAs a consequence the authors prove that in the Cayley graph \(\Gamma \big (B_{n}/Z(B_{n})\big )\) of any braid group \(B_{n}\), equipped with Garside's generating set, the axes of all pseudo-Anosov braids are strongly contracting.