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.