Agol-Groves-Manning’s Theorem predicts that, for every word-hyperbolic group we can easily construct, every quasiconvex subgroup is separable (otherwise, we would find a non-residually finite hyperbolic group!).

In this section, we use graphs of groups to build new hyperbolic groups:

Combination Theorem (Bestvina & Feighn): If $H$ is a malnormal subgroup of hyperbolic groups $G_1, G_2$, then $G_1\ast_H G_2$ is hyperbolic.

Recall: $H$ is called a malnormal subgroup of $G$ if it satisfies: if $gHg^{-1}\cap H\neq 1$, then $g\in H$.

For a proof, see M. Bestvina and M. Feighn, “A combination theorem for negatively curved groups”, J. Differential Geom., 35 (1992), 85–101.

Example: Let $F$ be free, $w\in F$ not a proper power. By Lemma 11, $\langle w\rangle\leq F$ is malnormal, so $D:=F\ast_{\langle w\rangle} F$ is hyperbolic. As a special case, if $\Sigma$ is closed surface of even genus $n=2k$, considered as the connected sum of two copies of the closed surface of genus $k$, then by Seifert-van Kampen Theorem, $\pi_1(\Sigma)=F_{2k}\ast_{\langle w\rangle} F_{2k}$ for some $w\in F_{2k}$.

Question: (a) Which subgroups of $D$ are quasiconvex? (b) Which subgroups of $D$ are separable?

We will start by trying to answer (b). The following is an outline of the argument: Let $\Gamma$ be a finite graph so that $\pi_1(\Gamma)=F$, let $\Gamma_{\pm}$ be two copies of $\Gamma$. Realize $w\in F$ as  maps $\partial^{\pm}: C\rightarrow \Gamma_{\pm}$, where $C\simeq S^1$. Let $X$ be the graph of spaces with vertex spaces $\Gamma_{\pm}$, edge space $C$, and edge maps $\partial^{\pm}$. Then clearly, $D\simeq \pi_1(X)$, and finitely generated subgroups $H\leq D$ are in correspondence with covering spaces $X^H\rightarrow X$. We can then use similar technique to sections 27 and 28.

Let us now make a few remarks about  elevations of loops. Let $f: C\rightarrow X$ be a loop in some space $X$, i.e., $C\simeq S^1$ and $\pi_1(C)\simeq\mathbf{Z}$. Consider an elevation of $f$:

The conjugacy classes of subgroups of $\mathbf{Z}$ are naturally in bijection with $\mathbf{N}\cup\{\infty\}$. The degree of the elevation is equal to the degree of the covering map $C'\rightarrow C$.

Definition: Suppose $X'\rightarrow X$ is a covering map and $\widehat{X}$ is an intermediate covering space, i.e., $X'\rightarrow X$ factors through $\widehat{X}\rightarrow X$, and we have a diagram

If $f'$ and $\widehat{f}$ are elevations of $f$ and the diagram commutes, then we say that $f'$ descends to $\widehat{f}$.

Let $\Gamma$  be a finite graph, $H\leq \pi_1(\Gamma)$ a finitely generated subgroup and $f: C\rightarrow\Gamma$ a loop. Let $\Gamma^H\rightarrow\Gamma$ be a covering space corresponding to $H$.

Lemma 29: Consider a finite collection of elevations $\{f'_i: C'_i\rightarrow\Gamma^H\}$ of $f$ to $\Gamma^H$, each of infinite degree. Let $\Delta\leq\Gamma^H$ be compact. Then for all sufficiently large $d>0$, there exists an intermediate, finite-sheeted covering space $\Gamma_d\rightarrow\Gamma$ satisfying: (a) $\Delta$ embeds in $\Gamma_d$; (b) every $f'_i$ descends to an elevation $\widehat{f}_i: \widehat{C}_i\rightarrow\Gamma_d$ of degree exactly $d$; (c) these $\widehat{f}_i$ are pairwise distinct.