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Let $\Sigma_{g}$ be a closed orientable surface of genus $g$. Let $d_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma_g$. For $g=1$, the mapping class group is $SL(2, \mathbb{Z})$ so $d_g=2$. For $g=2$, this paper proves that $d_g \leq 64$.

  • For what $g \geq 3$ (if any) is it known that $d_g$ is finite? This question has been asked here 3 years ago; maybe something has changed since then.
  • Is it known that $d_2=64$?
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There has been no progress on this question for many years. In particular, the precise value of $d_2$ is not known and it is not known if $d_g$ is finite for any $g \geq 3$.

The only related paper is this one by Korkmaz, which proves that any representation of the genus $g$ mapping class group to $GL(n,\mathbb{C})$ has abelian image for $n \leq 2g-1$ (this slightly improves results of Funar and Franks-Handel). Of course, for $n=2g$ there is the classical symplectic representation.

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