This is the second of three posts based on a guest lecture by Sam Kim.
Define to be the word that alternates between the letters and and has length . Suppose is a simplicial graph with a labeling of its edges . The associated Artin group is
For example, the braid group on strands has presentation and so is when is a two vertices connected by a singe edge and assigns the label to that edge.
The Coxeter group is the quotient of arising on adding the relations for all .
Gordon, Long and Reid were the first to consider the question of which Artin groups contain hyperbolic surface subgroups. Let be the triangle with edges labeled by . They found that most Artin groups of finite and Euclidean type admit hyperbolic surface subgroups. To be precise, they left the question unresolved for Artin groups associated to , , and , and amongst all others of finite and Euclidean type only the groups in the families , and fail to contain hyperbolic surface groups.
is type and is of finite type, is of type , and is of type and so of Euclidean type. For the first two, the problem remains open. We will discuss S. Kim’s (unpublished) account of why the last one admits a hyperbolic surface subgroup.
The standard presentation of a braid group (which you can find here) gives us that
First we show that this group contains a hyperbolic surface group. Let where is the figure-eight knot. is known to contain a hyperbolic surface group (see [Cooper-Long-Reid 1997]), and to admit the presentation
Using a similar argument to Mangum and Shanahan’s, one can embed into via , and . So also contains a surface group.
Now consider the subgroup that fixes the th string. It is generated by , and it admits the presentation of in these generators. It is an index subgroup of , so it also contains a surface group.
On the other hand, we can get an isomorphism , by fattening the th string (see Kent-Peifer2002]). We can visualize the th (fattened) puncture as the inner hole of the annulus, and the punctures , and arranged in cyclic order around the annulus, and invariant under a rotation of . Then the element corresponds, as a braid in , to a rotation of angle on the target annulus. This gives a decomposition of as a semi-direct product
where the first factor is generated by . Consider the index subgroup
It contains a surface subgroup, since it has finite index in . But since it is a direct product, the surface subgroup must project isomorphically into the first factor (since a hyperbolic surface group cannot contain ).
Next we will consider a different index–3 subgroup of ; namely, we take the subgroup of braids that fix the st and th punctures.
Let be the morphism given by the winding number of the st string around the th string. Then , where is a pair of pants (a sphere minus three disks). This can be seen by fattening the st and th strings, and observing that in a braid in , they do not wind around each other.
In his thesis, Davide Moroni proved that this exact sequence splits, giving . Thus, by the previous argument, contains a surface group.
Moroni also proved that , and this finishes the proof.