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 is a quasiconvex malnormal subgroup of hyperbolic groups , then is hyperbolic.

Recall: is called a malnormal subgroup of if it satisfies: if , then .

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 be free, not a proper power. By Lemma 11, is malnormal, so is hyperbolic. As a special case, if is closed surface of even genus , considered as the connected sum of two copies of the closed surface of genus , then by Seifert-van Kampen Theorem, for some .

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

We will start by trying to answer (b). The following is an outline of the argument: Let be a finite graph so that , let be two copies of . Realize as maps , where . Let be the graph of spaces with vertex spaces , edge space , and edge maps . Then clearly, , and finitely generated subgroups are in correspondence with covering spaces . We can then use similar technique to sections 27 and 28.

Let us now make a few remarks about elevations of loops. Let be a loop in some space , i.e., and . Consider an elevation of :

The conjugacy classes of subgroups of are naturally in bijection with . The degree of the elevation is equal to the degree of the covering map .

**Definition:** Suppose is a covering map and is an intermediate covering space, i.e., factors through , and we have a diagram

If and are elevations of and the diagram commutes, then we say that descends to .

Let be a finite graph, a finitely generated subgroup and a loop. Let be a covering space corresponding to .

**Lemma 29:** Consider a finite collection of elevations of to , each of infinite degree. Let be compact. Then for all sufficiently large , there exists an intermediate, finite-sheeted covering space satisfying: (a) embeds in ; (b) every descends to an elevation of degree exactly ; (c) these are pairwise distinct.

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5 December 2016 at 11.48 am

Giles GardamNote that the statement of the Combination Theorem given above isn’t quite right (although it works for free groups since they’re locally quasiconvex). See the discussion at http://mathoverflow.net/questions/140310/amalgmated-free-product-of-hyperbolic-groups-with-one-malnormal-and-one-virtual