Periodic Elements in Finite-Type Artin–Tits Groups and Stability Conditions

Periodic elements in finite-type Artin–Tits groups are elements some positive power of which is central. We give a dynamical characterisation of periodic elements via their action on the corresponding 2-Calabi–Yau category and on its space of (fusion equivariant) Bridgeland stability conditions. The main theorem is that an element β is periodic if and only if β has a fixed point in the stability manifold.