We prove that any `finite-type' component of a stability space of a
triangulated category is contractible. The motivating example of such a
component is the stability space of the Calabi--Yau-N category
D(ΓNQ) associated to an ADE Dynkin quiver. In addition to
showing that this is contractible we prove that the braid group
Br(Q) acts freely upon it by spherical twists, in particular
that the spherical twist group Br(ΓNQ) is isomorphic to
Br(Q). This generalises Brav-Thomas' result for the N=2
case. Other classes of triangulated categories with finite-type components in
their stability spaces include locally-finite triangulated categories with
finite rank Grothendieck group and discrete derived categories of finite global
dimension.Comment: Final version, to appear in Geom. Topo