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:

diagram

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

diagram1

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.

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