On mapping class group quotients by powers of Dehn twists and their representations

TL;DR

This paper surveys known results on quotients of mapping class groups by powers of Dehn twists, focusing on their finite-dimensional representations, and discusses new connections and methods for constructing such representations.

Contribution

It reveals that in genus 2, the Fibonacci TQFT representation is a specialization of the Jones representation and discusses a method for generating large families of representations.

Findings

Fibonacci TQFT representation is a specialization of the Jones representation in genus 2.

Finite quotients can be constructed from Zariski dense representations into semisimple Lie groups.

Long and Moody's method provides a way to generate large families of mapping class group representations.

Abstract

The aim of this paper is to survey some known results about mapping class group quotients by powers of Dehn twists, related to their finite dimensional representations and to state some open questions. One can construct finite quotients of them, out of representations with Zariski dense images into semisimple Lie groups. We show that, in genus 2, the Fibonacci TQFT representation is actually a specialization of the Jones representation. Eventually, we explain a method of Long and Moody which provides large families of mapping class group representations.