Recent developments in the theory of stability conditions and its relation to Teichmuller
theory have revealed a deep connection between triangulated categories and surfaces.
Motivated by this, we prove a categorical analogue of the Nielsen-Thurston classification
theorem for the rank two generalised braid groups by viewing them as (sub) groups of
autoequivalences of certain triangulated categories. This can be seen as a (categorical)
generalisation of the classification known for the type A braid groups when viewed as …