Categorification and Dynamics in Generalised Braid Groups

TL;DR

This paper categorifies generalized braid groups as autoequivalence groups of triangulated categories, classifies their elements using a Nielsen-Thurston type theorem, and links their dynamics to categorical entropy and matrix computations.

Contribution

It provides a categorical analogue of the Nielsen-Thurston classification for rank two generalized braid groups, extending known results to non-simply-laced types and establishing computable invariants.

Findings

Categorical actions categorify Burau representations.

Classification of elements via an algorithm based on mass growth.

Pseudo-Anosov elements' entropy is computable from rank two matrices.

Abstract

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 mapping class groups of the punctured discs. Firstly, we realise the generalised braid groups as groups of autoequivalences through categorical actions that categorify the corresponding Burau representations. These categorifications are achieved by constructing certain algebra objects in the tensor categories associated to the quantum group $U_q(\mathfrak{sl}_2)$, generalising the construction of zigzag algebras used in the categorical actions of simply-laced-type braid groups to include the non-simply-laced-types. By viewing the elements of the generalised braid groups as autoequivalences of triangulated categories, we study their dynamics through mass growth (categorical entropy), as introduced by Dimitrov--Haiden--Katzarkov--Kontsevich. Our classification is then achieved in a similar fashion to Bestvina-Handel's approach to the Nielsen-Thurston classification for mapping class groups. Namely, our classification can be effectively decided through a given algorithm that also computes the mass growth of the group elements. Moreover, it shows that the mass growth of the pseudo-Anosov elements are computable from certain rank two matrices. This is the author's PhD thesis.