# Presentation for an infinite index subgroup of the braid group

If $H$ is an infinite index subgroup of the braid group $\mathcal{B}_n$, is there a way to find a presentation for $H$ ?

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Thanks Mathew. Maybe the braid group is not an easier route after all. To start with, $H$ is actually a subgroup of $Mod(S)$ generated by three or more Dehn twists, where $S$ is a compact, orientable surface with nonempty boundary. $H$ also happens to be a subgroup of the braid group $\mathcal{B}_n$. The braid and disjointness relations are obvious for the generating Dehn twists of $H$. It is not clear, however, whether these are all the defining relations. –  Jim B Apr 30 '10 at 1:23
Do you know them as words in standard generators? Perhaps you could experiment a bit with Magma or Gap using a known presentation. –  Matthew Stover Apr 30 '10 at 1:32
Thanks Mathew! GAP is not very useful to me. Based on your suggestion, I've played with Magma and got some interesting results. I still couldn't find the presentation I want though. –  Jim B May 4 '10 at 3:43
Your question isn't well-posed. The group multiplication table is a presentation of any group. Are you ruling out things like that, and if so, how? –  Ryan Budney Jul 5 '12 at 6:25

Unless you have a specific type of subgroup, like one that acts cocompactly on something, you're in deep trouble. Braid groups are incoherent (see Artin groups, 3-manifolds and coherence by Cameron Gordon). That is, there are finitely generated subgroups that are not finitely presented.

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there are also infinitely generated subgroups. –  Ian Agol Jul 5 '12 at 14:33
If the subsurface is planar, then the subgroup generated by Dehn twists about the curves will be a subgroup of a central extension of the braid group. If your curves are labelled $c_1, c_2, \ldots, c_k$, such that $c_i\cap c_j=\emptyset$ if $|i-j|>1$, and $|c_i\cap c_{i+1}|=2$, and the subsurface is planar, then the group will generate the mapping class of the planar subsurface, since these are the form of the standard generators for the braid group (the central extension comes from Dehn twists around boundary components).