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Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of Agol-Groves-Manning, and generalizes some earlier work they did a few years ago. Jason referred to the main theorem during his talk as the “Goal Theorem” (I guess it was the goal of his lecture), but I’m going to call it the Weak Separation Theorem, since that is a somewhat more descriptive name. The statement of the theorem is as follows.

Weak Separation Theorem (Agol-Groves-Manning): Let G be a hyperbolic group, let H be a subgroup of G which is quasiconvex, and isomorphic to the fundamental group of a virtually special NPC cube complex, and let g be an element of G which is not contained in H. Then there is a surjection $\phi:G \to \bar{G}$ so that

1. $\bar{G}$ is hyperbolic;
2. $\phi(H)$ is finite; and
3. $\phi(g)$ is not contained in $\phi(H)$.

In the remainder of this post I will try to explain the proof of this theorem, to the extent that I understand it. Basically, this amounts to my summarizing Manning’s talk (or the part of it that I managed to get down in my notes); again, any errors, foolishness, silly blog post titles etc. are due to me.