Two questions on mapping class groups

TL;DR

This paper investigates the residual finiteness of central extensions of mapping class groups and estimates the largest SU(N) groups into which these groups can map with finite images, using quantum representations.

Contribution

It proves residual finiteness of certain central extensions of mapping class groups and provides bounds on the size of SU(N) for finite image homomorphisms, based on quantum representations.

Findings

Central extensions of $M_g$ by $ extbf{Z}$ are residually finite.

Homomorphisms from $M_g$ to $SU(N)$ have finite image for specific N.

Homomorphisms into $SL([ ext{sqrt}(g+1)], extbf{C})$ are finite.

Abstract

We show that central extensions of the mapping class group $M_g$ of the closed orientable surface of genus $g$ by $\Z$ are residually finite. Further we give rough estimates of the largest $N=N_g$ such that homomorphisms from $M_g$ to SU(N) have finite image. In particular, homomorphisms of $M_g$ into $SL([\sqrt{g+1}],\C)$ have finite image. Both results come from properties of quantum representations of mapping class groups.