Contractible stability spaces and faithful braid group actions

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 ( Γ N Q ) 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 ( Γ N Q ) is isomorphic to Br ( Q ) . This generalises the result of Brav–Thomas 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