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](https://arxiv.org/abs/math/0010310) proves that $d\_g \leq 64$. 


* For what $g \geq 3$ (if any) is it known that $d\_g$ is finite? [This question](https://mathoverflow.net/questions/216208/current-status-of-the-linearity-of-mapping-class-group) has been asked here 3 years ago; maybe something has changed since then.
* Is it known that $d\_2=64$?



Minimum dimension of faithful representation of mapping class groups?
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](https://arxiv.org/abs/1104.4816) 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.


