Periodic elements in finite type Artin-Tits groups and stability conditions

TL;DR

This paper characterizes periodic elements in finite type Artin-Tits groups through their action on associated 2-Calabi--Yau categories and stability conditions, linking algebraic properties with geometric dynamics.

Contribution

It provides a dynamical characterization of periodic elements via their fixed points in the stability manifold, connecting group theory and category theory.

Findings

Periodic elements correspond to fixed points in the stability manifold.

A dynamical criterion for periodicity in Artin-Tits groups.

Bridgeland stability conditions relate to algebraic periodicity.

Abstract

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 $\beta$ is periodic if and only if $\beta$ has a fixed point in the stability manifold.