Last time: Theorem 21 (Groves–Manning–Osin): If is hyperbolic rel then there exists a finite subset such that if then
(a) is injective;
(b) is hyperbolic rel .
Theorem 22 (Gromov, Olshanshkii, Delzant): If is hyperbolic relative to the infinite cyclic then there is a such that for all there exists a hyperbolic such that for each .
The proof is an easy application of Groves–Manning–Osin.
Definition: If (infinite cyclic) is malnormal then we say are independent. A group G is omnipotent if for every independent there exists a such that for all there exists a homomorphism $\phi$ from to a finite group such that for all .
Omnipotence strengthens residual finiteness for torsionfree groups.
Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.
We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.
Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup of any hyperbolic group is separable.
Let . The idea is to Dehn fill to get a new hyperbolic group in which the image is finite and . If we could do this, we would be done by residual finiteness. This works if is malnormal. But it probably isn’t. Fortunately, we can quantify how far is from being malnormal:
Definition: The height of is the maximal such that there are distinct cosets such that the intersection
H is height iff is finite. In a torsionfree group, is height iff is malnormal.
Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.
Agol, Groves and Manning are able to prove:
Theorem 24: Let be a (torsionfree) residually finite hyperbolic group, and a quasiconvex subgroup of height . Let . Then is an epimorphism to a hyperbolic group such that
(i) is quasiconvex in ;
(iii) has height .
The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of . Theorem 22 is an easy consequence.