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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

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  • $\begingroup$ Are genus $g\gg 0$ mapping class groups known to have any faithful representations? Just curious. $\endgroup$
    – Donu Arapura
    Commented Apr 7, 2012 at 23:02
  • $\begingroup$ The mapping class group of $S^1 \times S^1$ is isomorphic to $GL_2\mathbb Z$. The mapping class group of a genus $2$ surface has a faithful linear representation of dimension $64$ -- this is because the genus $2$ mapping class group is isomorphic to the hyper-elliptic subgroup, so closely related to braid groups of the sphere. arxiv.org/abs/math/0010310 $\endgroup$
    – Ryan Budney
    Commented Apr 7, 2012 at 23:07
  • $\begingroup$ Thanks for the reference to genus two. So I take it that $g>2$ is open? $\endgroup$
    – Donu Arapura
    Commented Apr 7, 2012 at 23:21
  • $\begingroup$ As far as I know. Compact surface mapping class groups are conjectured to be all linear, but nobody knows of a good candidate representation for mapping class groups that they hope is faithful. $\endgroup$
    – Ryan Budney
    Commented Apr 7, 2012 at 23:44

1 Answer 1

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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.

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