Last time: Theorem 21 (Groves–Manning–Osin): If G is hyperbolic rel \mathcal P then there exists a finite subset A \subseteq G\setminus 1 such that if \bigcup_i N_i \cap A = \emptyset then
(a) P_i/N_i \to G/\mathcal N is injective;
(b) G is hyperbolic rel P_i/N_i.

Theorem 22 (Gromov, Olshanshkii, Delzant): If G is hyperbolic relative to the infinite cyclic \{\langle g_1\rangle,\dots,\langle g_n \rangle\} then there is a K>0 such that for all l_1,\dots,l_n>0 there exists a \phi : G \to G' hyperbolic such that o(\phi(g_i))=Kl_i for each i.

The proof is an easy application of Groves–Manning–Osin.

Definition: If \{\langle g_1 \rangle,\dots,\langle g_n\rangle\} (infinite cyclic) is malnormal then we say g_1,\dots,g_n are independent. A group G is omnipotent if for every independent g_1,\dots,g_n there exists a K>0 such that for all l_1,\dots,l_n>0 there exists a homomorphism $\phi$ from G to a finite group such that o(\phi(g_i)) = Kl_i for all i.

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 H of any hyperbolic group G is separable.

Let g \in G \setminus H. The idea is to Dehn fill H to get a new hyperbolic group \bar G in which the image \bar H is finite and \bar g \not\in\bar H. If we could do this, we would be done by residual finiteness. This works if H is malnormal. But it probably isn’t. Fortunately, we can quantify how far H is from being malnormal:

Definition: The height of H is the maximal n \in \mathbb N such that there are distinct cosets g_1 H,\dots,g_n H \in G/H such that the intersection
g_1 H g_1^{-1} \cap \dots \cap g_n H g_n^{-1}
is infinite.

H is height 0 iff H is finite. In a torsionfree group, H is height 1 iff H 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 G be a (torsionfree) residually finite hyperbolic group, and H a quasiconvex subgroup of height k. Let g \in G\setminus H. Then is an epimorphism \eta: G \to \bar G to a hyperbolic group such that
(i) \eta(H) is quasiconvex in \bar G;
(ii) \eta(g) \not\in\eta(H);
(iii) \eta(H) has height \leq k-1.

The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of H. Theorem 22 is an easy consequence.

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