Lower bounds on dimensions of faithful representations of braid groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:50:29Z http://mathoverflow.net/feeds/question/92975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92975/lower-bounds-on-dimensions-of-faithful-representations-of-braid-groups Lower bounds on dimensions of faithful representations of braid groups Ryan Budney 2012-04-03T08:26:45Z 2012-04-07T21:59:55Z <p>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</p> <p>$$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}]$$</p> <p>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. </p> <p>My question for the community is, what kind of <em>lower bounds</em> are known on $k$ for there to be a faithful representation</p> <p>$$B_n \to GL_k \mathbb C$$ </p> <p>Also, are there any analogous bounds for the same question but for mapping class groups? </p> <p>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? </p> <p>edit: Marin has a lower bound $k \geq n+1$ provided $n$ is sufficiently large. </p> <p>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 $n \leq 3g-3$. <a href="http://arxiv.org/abs/1108.3241" rel="nofollow">http://arxiv.org/abs/1108.3241</a></p> http://mathoverflow.net/questions/92975/lower-bounds-on-dimensions-of-faithful-representations-of-braid-groups/93029#93029 Answer by Andy Putman for Lower bounds on dimensions of faithful representations of braid groups Andy Putman 2012-04-03T18:58:07Z 2012-04-03T18:58:07Z <p>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 <a href="http://arxiv.org/abs/1104.4816" rel="nofollow">http://arxiv.org/abs/1104.4816</a>.</p>