Contractible stability spaces and faithful braid group actions

Abstract

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 $\mathcal{D}({\Gamma }_{N}Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\mathrm{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\mathrm{Br}({\Gamma }_{N}Q)$ is isomorphic to $\mathrm{Br}(Q)$. 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.

Final version, to appear in Geom. Topol

References in corpus (1)

• Decorated marked surfaces: spherical twists versus braid twists

Cited by in corpus (3)

• On pseudo-Anosov autoequivalences
• Cluster exchange groupoids and framed quadratic differentials
• Finite presentations for spherical/braid twist groups from decorated marked surfaces