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

is infinite.

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 ;

(ii) ;

(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.

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