Curve complexes and Garside groups

We present a simple construction which associates to every Garside group a metric space, called the additional length complex, on which the group acts. These spaces share important features with curve complexes: they are $\delta$-hyperbolic, infinite, and typically locally infinite graphs. We conjecture that, apart from obvious counterexamples, additional length complexes are always of infinite diameter. We prove this conjecture for the classical example of braid groups $(B_n,\mathrm{\Delta })$; moreover, in this framework, reducible and periodic braids act elliptically, and at least some pseudo-Anosov braids act loxodromically. We conjecture that for $B_n$, the additional length complex is actually quasi-isometric to the curve complex of the $n$ times punctured disk.