The Torelli group is defined as the kernel of the symplectic representation
It is known to be finitely generated, but the question of finite presentation is still open. Given that the mapping class group is well known to be finitely presented, a natural question to ask is then, under what conditions, if any, does every finitely generated subgroup of a given group permit a finite presentation? A group satisfying this property is said to be coherent. Of course if the mapping class group could be shown to be coherent then the question of finite presentation of the Torelli group is trivially solved. All I have done is said; if every finitely generated subgroup of the mapping class group is finitely presented, then the Torelli group is finitely presented, which isn’t very helpful. But the question of coherence could provide a roundabout way of solving this problem. More importantly however it serves as motivation for the question; under what conditions can a group be said to be coherent.
Let’s show that some groups are coherent and that some groups aren’t, to make sure that there is actually a question here:
Example: Some non-coherent groups
Let be the homomorphism sending the generators of one copy of , and of the other copy all to 1. Then is finitely generated, but not finitely presented, with presentation where . Details of this can be found here. This not only serves as an example of an incoherent group, but also provides a tool for showing other groups are not coherent, simply by embedding copies of in them. For instance, we can embed in as it is generated by
Using this we can embed in with generators , , and . And similarly we can embed in with . So we have that these linear groups are also not coherent.
Example: Surface groups are coherent
Surface groups are 1-relator groups. Moreover, any subgroup of a surface group is another surface group, and thus is finitely presented. This is a nice easy case as surfaces have a powerful underlying topological restriction, namely the classification of surfaces.
In fact it is also known that the fundamental groups of 3-manifolds are coherent, and similarly it is the underlying topological properties that allow this to be shown, specifically the Scott core theorem.