  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 $(B_n,\Delta)$; 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.
