Burau representations and the mapping class group of the twice-punctured torus

Grace Garden
Sorbonne Université

Abstract

The twice-punctured torus S_(1,2) occupies an intermediate position between surface mapping class groups, braid groups, and surface groups. In this talk, we use these connections to study representations from the mapping class group of the twice-punctured torus PMod(S_(1,2)) to SL(3, C). A particular focus is placed on those coming from the Burau representations, a famous family of representations classically defined on braid groups. Our main result is a classification of representations from PMod(S_(1,2)) to SL(3, C) up to conjugacy. If time permits, we will also give a survey on specific parameters for Burau representations with geometric meaning. This is joint work with Elisha Falbel and Antonin Guilloux.