Question

Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation

$$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}] $$

i.e. the range is the group of invertible matrices of rank $n \choose 2$ whose entries are from a Laurent polynomial ring over the integers.

My question for the community is, what kind of lower bounds are known on $k$ for there to be a faithful representation

$$B_n \to GL_k \mathbb C$$

Also, are there any analogous bounds for the same question but for mapping class groups?

I have a vague recollection that there are some good answers to this question but I forget who they're due to. I also strongly suspect if you restrict to unitary representations there are some stronger results, maybe due to Stoimenow or Marin?

edit: Marin has a lower bound $k \geq n+1$ provided $n$ is sufficiently large.

edit2: Korkmaz has a theorem that says no representation from the genus $g$ mapping class group to $GL_n \mathbb C$ can be faithful, provided $n \leq 3g-3$. http://arxiv.org/abs/1108.3241

Answer 1

Slightly improving a theorem of Franks and Handel, Korkmaz proved that any representation from the genus $g$ mapping class group to $\text{GL}_n(\mathbb{C})$ is trivial if $n \leq 2g-1$. This bound is sharp because of the standard symplectic representation. See http://arxiv.org/abs/1104.4816.