Two questions on mapping class groups

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.

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