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 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.