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Last time: Theorem 21 (Groves–Manning–Osin): If is hyperbolic rel then there exists a finite subset such that if then
(a) is injective;
(b) is hyperbolic rel .
Theorem 22 (Gromov, Olshanshkii, Delzant): If is hyperbolic relative to the infinite cyclic then there is a such that for all there exists a hyperbolic such that for each .
The proof is an easy application of Groves–Manning–Osin.
Definition: If (infinite cyclic) is malnormal then we say are independent. A group G is omnipotent if for every independent there exists a such that for all there exists a homomorphism $\phi$ from to a finite group such that for all .
Omnipotence strengthens residual finiteness for torsionfree groups.
Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.
We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.
Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup of any hyperbolic group is separable.
Let . The idea is to Dehn fill to get a new hyperbolic group in which the image is finite and . If we could do this, we would be done by residual finiteness. This works if is malnormal. But it probably isn’t. Fortunately, we can quantify how far is from being malnormal:
Definition: The height of is the maximal such that there are distinct cosets such that the intersection
H is height iff is finite. In a torsionfree group, is height iff is malnormal.
Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.
Agol, Groves and Manning are able to prove:
Theorem 24: Let be a (torsionfree) residually finite hyperbolic group, and a quasiconvex subgroup of height . Let . Then is an epimorphism to a hyperbolic group such that
(i) is quasiconvex in ;
(iii) has height .
The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of . Theorem 22 is an easy consequence.
Lemma 29: Suppose is a finite set of infinite degree elvations and is compact. Then for all sufficiently large , there exists an intermediate covering such that
(a) embeds in
(b) every descends to an elevation of degree
(c) the are pairwise distinct
Proof: We claim that the images of never share an infinite ray (a ray is an isometric embedding of ). Neither do two ends of the same elevation . Let’s claim by passing to the universal cover of , a tree .
For each , lift to a map . If and share an infinite ray then there exists such that and overlay in an infinite ray. The point is that correspond to cosets and . But this implies that
This implies that . So . A similar argument implies that the two ends of do not overlap in an infinite ray. This proves the claim.
Let be the core of . Enlarging if necessary, we can assume that
(ii) is a connected subgraph;
(iii) for each , for some , ;
(iv) for each , .
For each identifying with so that is identified with and is identified with . Let
For all sufficiently large ,
Now, the restriction of factors through , where . This is a finite-to-one immersion, so, by theorem 5, we can complete it to a finite-sheeted covering map as required.
Theorem 19: is LERF.
Recall the set-up from the previous lecture. We built a graph of spaces for .
Proof: Let be finitely generated. Let be the corresponding covering space of and let be compact. Because is finitely generated, there exists a subgraph of spaces such that . We can take large enough so that . We can enlarge so that it contains every finite-degree edge space of . Also enlarge so that
for any . For each let and let incident edge map of infinite degree .
Applying lemma 29 to , for some large , set . (Here we use the fact that vertex groups of are finitely generated)
Define as follows:
For each , the edge space is the that the lemma produced from the corresponding .
Now, by construction, can be completed to a graph of spaces so that the map
factors through and embeds. Let be identical to except with +’s and -‘s exchanged. Clearly satisfies Stallings condition, as required.
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.