Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of Agol-Groves-Manning, and generalizes some earlier work they did a few years ago. Jason referred to the main theorem during his talk as the “Goal Theorem” (I guess it was the goal of his lecture), but I’m going to call it the Weak Separation Theorem, since that is a somewhat more descriptive name. The statement of the theorem is as follows.
Weak Separation Theorem (Agol-Groves-Manning): Let G be a hyperbolic group, let H be a subgroup of G which is quasiconvex, and isomorphic to the fundamental group of a virtually special NPC cube complex, and let g be an element of G which is not contained in H. Then there is a surjection so that
- is hyperbolic;
- is finite; and
- is not contained in .
In the remainder of this post I will try to explain the proof of this theorem, to the extent that I understand it. Basically, this amounts to my summarizing Manning’s talk (or the part of it that I managed to get down in my notes); again, any errors, foolishness, silly blog post titles etc. are due to me.
Recall from my previous post that an NPC (non-positively curved) cube complex is a compact quotient of a CAT(0) cube complex by a group (in this case H) acting properly discontinuously, and that the complex is virtually special if it has a finite (orbi-)cover satisfying the Haglund-Wise conditions (i.e. hyperplanes are embedded, two-sided, and there are no self- or interosculations). The fundamental group of a virtually special cube complex is a linear group (i.e. embeds in for some n), and is therefore residually finite; this means that the intersection of all finite index normal subgroups consists only of the identity element. The fact that all finitely generated linear groups are residually finite is known (to topologists, anyway) as Selberg’s Lemma. Roughly, the idea is to consider the ring A of matrix entries in a faithful linear representation, and then the desired finite index subgroups are the kernels of maps to for suitable prime ideals in (see e.g. here for more details). More generally, if G is a group, a subgroup H is said to be separable if for any g not in H there is a homomorphism from G to a finite group so that the image of g is disjoint from the image of H. Groups in which every finitely generated subgroup is separable are said to be LERF; it is a consequence of Agol’s proof of the VHC that every hyperbolic 3-manifold group is LERF, but we are getting ahead of ourselves here.
The Weak Separation Theorem is in the direction of showing that the subgroup H is separable; if the quotient could be taken to be finite, that is exactly what it would show. But finding a quotient where is finite turns out to be good enough for Agol’s purposes.
An important special case of the theorem is when H is almost malnormal. A subgroup H of G is normal if conjugation in G sends H to itself. H is malnormal if for all , where superscript denotes conjugation, and H is almost malnormal if is finite for all .
Bowditch showed that if G is hyperbolic and H is quasiconvex and almost malnormal in G, then the pair is relatively hyperbolic. The concept of relative hyperbolicity generalizes the fundamental group of a noncompact complete negatively curved manifold of finite volume; the fundamental group is not (necessarily) a hyperbolic group (although it is in some cases!) but the lack of hyperbolicity is concentrated in the noncompact cusp of the manifold; the fundamental group of the cusp itself might or might not be hyperbolic. It is a parabolic subgroup of the fundamental group, and the pair is relatively hyperbolic. Abstractly, a pair is relatively hyperbolic, where is a collection of conjugacy classes of subgroups of , if the space obtained by attaching “horoballs” to the conjugates of the subgroups in in (the Cayley graph of) G is hyperbolic (see here for more details). Note that if the subgroups are themselves hyperbolic, then is also hyperbolic (in the absolute sense).
Relatively hyperbolic groups invite relatively hyperbolic Dehn filling, by analogy with Thurston’s hyperbolic Dehn surgery for (noncompact) hyperbolic 3-manifolds with cusps. Suppose is relatively hyperbolic, where . For each i choose some normal subgroup of . The quotient of by the normal closure (in G) of all the is called a filling of G by the , and is denoted . If each of the is finite index in , we call it a peripherally finite filling. The fundamental theorem of hyperbolic Dehn surgery, due (in this form) originally to Osin, is as follows:
Theorem (Osin): Let F be a finite subset of G, and let be relatively hyperbolic. Then there is a finite subset B of G, so that for any filling for which B does not intersect any of the , one has the following:
- for each i;
- the image pair in the quotient is relatively hyperbolic
- the restriction of to is injective.
- for all i
Actually, the fourth condition is not proved by Osin, but can be deduced with “a bit of work”, according to Jason. Notice that if the filling is peripherally finite, so that all are finite, then these quotients of the parabolic groups are trivially hyperbolic (since every finite group is hyperbolic), and therefore the quotient group is also hyperbolic.
If H is almost malnormal, Osin’s surgery theorem implies the Weak Separability Theorem, as follows. Since H is almost malnormal and quasiconvex, and G hyperbolic, by Bowditch the pair is relatively hyperbolic. We let F (as in the statement of Osin’s theorem) consist only of the element , and let be the finite set of “bad” fillings the theorem guarantees. Since H is assumed to be virtually special, it is residually finite, and therefore contains a finite index normal subgroup N missing B. Taking the quotient of G by the normal closure of N gives a surjection to a group in which the image of H is finite, and disjoint from the image of g, as desired.
What if H is not malnormal? Then one must induct on an invariant called the height, which measures the failure of the group to be almost malnormal. The idea of height was introduced by Gitik-Mitra-Rips-Sageev.
Definition: If H is quasiconvex in a hyperbolic group G, the height of H is the least integer n so that if there are elements so that are distinct, then the intersection of conjugates is finite.
H has height 0 if and only if it is finite. It has height 1 if and only if it is almost malnormal and infinite. One can define a complex whose k-simplices are the -fold infinite intersections of distinct conjugates of H, and height is then the dimension of this complex plus one. Gitik-Mitra-Rips-Sageev prove that every quasiconvex subgroup of a hyperbolic group has finite height. They also show that for any k there are only finitely many H-conjugacy classes of infinite groups of the form (this is vacuous for k as big as the height or bigger). The minimal such infinite intersections are not far from being malnormal, and after a suitable modification, will give rise to a suitable relatively hyperbolic family.
Start with , the collection of H-conjugacy classes of minimal infinite intersections of the form (these are conjugacy classes of subgroups in H). These subgroups are intersections of quasiconvex subgroups, and are easily seen to be quasiconvex themselves. Replace each D in by its commensurator in H (note that each D is finite index in ), and then choose one such subgroup per H-conjugacy class. This produces a new collection of conjugacy classes of subgroups in H. Observe that the elements of are almost malnormal — in fact is an almost malnormal collection, meaning that nontrivial conjugates of one subgroup intersect any other subgroup in the collection in a finite set — so that is relatively hyperbolic (Bowditch’s criterion for relative hyperbolicity generalizes to almost malnormal collections of quasiconvex subgroups).
Now, replace each D in by its commensurator in G; again each D is finite index in . This produces a collection of subgroups of G; choose one subgroup for each G-conjugacy class to arrive finally at an almost malnormal collection of quasiconvex subgroups of G, and deduce (by Bowditch again) that is relatively hyperbolic.
Definition: A filling with each in some in as above is an H-filling if whenever is infinite for some in , then is contained in D.
An H-filling by definition induces a filling of H, i.e. a quotient . With this terminology, Agol-Groves-Manning prove
Theorem (Agol-Groves-Manning): Let G be hyperbolic, let H be quasiconvex in G of height at least 1, and let and be as above, and let g be in . Then for all “sufficiently long” peripherally finite H-fillings ,
- is isomorphic to the induced filling of H;
- is quasiconvex in ;
- is not contained in ; and
- the height of is strictly less than the height of H.
Let’s not worry too much about what the condition “sufficiently long” means here; suffice it to say that such fillings can be found if H is residually finite.
Now, in the setup of the Weak Separation Theorem, the subgroup H is the fundamental group of a virtually special NPC complex, and is therefore residually finite. So we can apply this filling theorem of AGM to reduce the height of H. But now one is stuck, because the resulting image , while of strictly lower height, might not be residually finite. Here is where Wise’s Malnormal Special Quotient Theorem (alluded to at the end of my previous post) comes in. The statement of the MSQT is as follows:
Malnormal Special Quotient Theorem (Wise): Let H be hyperbolic, let be relatively hyperbolic, where the in are almost malnormal and quasiconvex. Suppose H is the fundamental group of a virtually special NPC complex. Then there are finite index subgroups in the so that if is any peripherally finite filling with contained in the , then is the fundamental group of a virtually special NPC complex.
The MSQT implies that, providing we are careful for our H-filling to kill subgroups contained in the , the image of H will be virtually compact special, and the induction can be continued, reducing the height of (the image of) H until the Weak Separation Theorem is proved.