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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.
We will see two examples of non-quasiconvex subgroups in this section. The first one is NOT a hyperbolic group, while the second one is.
Example: For the first example, let
with one eigenvalue (the larger one) . Notice that does not fix any non-zero vectors in (such a map is called Anosov).
Now let . This is a group. The group law works like this: for any , . Pick , . The map is, by the following analysis, NOT a quasi-embedding:
Choose such that . All norms on are bilipschitz, so there exists such that . Therefore, for sufficiently large , , and so . On the other side, we have . It follows that is not a quasi-embedding.
Example: For the second example, let be a hyperbolic surface. An automorphism of is called pseudo-Anosov if for any smooth closed curve on and any , is not homotopic to . Let be the mapping torus of , i.e., , with the relation generated by .
Under these assumptions, we are able to use a theorem of Thurston asserting that, must be a hyperbolic 3-manifold. (W. Thurston, “On the geometry and dynamics of diffeomorphisms of surfaces,” Bull. Amer. Math. Soc. vol 19 (1988), 417-431)
Hence, if is closed, then is also closed. So acts nicely on (actually ), and so is word-hyperbolic by the Švarc–Milnor Lemma. Then a similar argument to the previous shows the natural map is NOT a quasi-embedding.
For concrete examples, see A. Casson & S. Bleiler, “Automorphisms of Surfaces After Nielsen and Thurston”.
After the two examples, let us switch to a property for all hyperbolic groups:
Theorem 7: Hyperbolic groups are finitely presented.
In order to prove this theorem, we need the following lemma:
Lemma 9: Let be two geodesics in a -hyperbolic metric space , . (If is longer than , say, then extend by the constant map). Then for any , .
Proof: Case 1: there is such that . Without loss of generality, assume , then . So, .
Case 2: there is no such that . Then must be within distance of . Apply a similar argument to the previous, we see .
Proof of Theorem 7: Let be -hyperbolic, with the generating set . Let be any relation, which corresponds to a loop in the Cayley graph . We can always take with and geodesics in and , by “triangulating”.
Write , . Denote , , . An easy induction shows that
But Lemma 9 implies that for all , so we have written the loop as a product of conjugates of loops of length at most . Therefore, the set of all loops of length at most is a finite set of relations for .