(some images might appear soon)
Somehow I decided to wake up at 6:30 a.m. every Thursday to attend Bruce Kleiner‘s 9:30 course in NYU this semester. So far it’s been fun~
I learned about this thing called graph of groups. If you have been reading posts on this blog regarding any geometric group theory stuff (especially those posts related to Kleiner), then warning: this ‘graph’ has nothing to do with the Cayley graph. It’s not much about geometry but a rather ‘category-theoretical’ thing. Well, at this point you may think that you hate those algebra prople and is ready to leave…just don’t do that yet, because I hated them too, and now I finally got a tiny bit of understanding and appreciation on what those abstract non-sense was all about! :-P
So how do we connect cool 3-manifold stuff (incompressible surfaces, loops, embedded discs Heegaard splittings etc.) to groups?
Well, one handy thing is of course the Dehn’s lemma:
Theorem: For 3-manifold with boundary, if the inclusion map is not injective, then there exists a simple non-trivial loop in bounding an embedded disc in .
Note: Dehn’s theorem was proved by Papakyriakopoulos, I talked about it in this pervious post, although not exactly stated in this form, we can see that Dehn’s lemma follows easily from the loop theorem.
This means we can say things about the 3-manifold by only looking solely at maps between groups!
That’s cool, but sometimes we find groups and just one map between two groups are not enough, and that’s when graph of groups comes in:
Definition: A graph of groups is a graph with vertice set , edge set , to each vertex we associate a group and to each edge (say connecting ) we also associate a group , together with a pair of injective homomorphisms , .
In our context, we should think of this as gluing together a bunch of spaces and take the fundamental group of those spaces, along with their pairwise intersections, as our vertice and edge groups. Just note that we need to have injections from the edge group to vertice groups. For simplicity one may first restrict oneself to the case where all edge groups are trivial (say spaces glued along contractible spaces).
There is something called the fundamental group of a graph of groups which is essentially the fundamental group of the resulting space after you glued spaces according to the given graph of groups. Note that the injection associated to edges takes into account how gluing of different pairs interact with each other (that is to say, for example, on a homotopy level it knows about triple intersections, etc.)
Theorem: Any hyperbolic group (plus obvious conditions, namely torsion free and does not split over a finite cyclic group) with 1-dimensional boundary has homeomorphic to , the Sierpinski carpet or the Menger curve.
When , my wonderful advisor Dave Gabai proved that would act discrete and cocompactly on by isometries. (i.e. it’s almost the fundamental group of some hyperbolic surface except for possible finite order elements which make the action not properly discontinuous.)
Now the next step is of course figuring out when does groups act on , we have:
Cannon’s conjecture: If hyperbolic group has boundary , then acts discretely and cocompactly on by isometries.
This conjecture was also mentioned another pervious post. Turns our we do not know much about groups with boundary. However, using graph of groups, they were able to show:
Theorem: If Cannon’s conjecture is true, then those hyperbolic groups with Sierpinski carpet boundary are fundamental groups of hyperbolic 3-manifolds with totally geodesic boundary.
i.e. the idea is to ‘extend’ the group with Sierpinski carpet boundary to a group having sphere boundary. Of course as sets we can embed the carpet into a sphere and start to ‘reflect it along the boundary of the ‘holes’, continue the process and eventually the union of all copies of the carpets is the entire . The problem is how to ‘reflect’ a group?
First, since the boundary is homeomorphic to the carpet, there are countably many well-defined ‘boundary circles’, the group acts on the set of boundary circles. They showed this action has only finitely many different orbits. (those orbits of boundary circles will eventually correspond to those totally geodesic boundary components of our resulting 3-manifold). We pick one boundary circle from each orbit and denote their stabilizers each .
Define a graph of groups with two vertices both labeled , with edges, all going from one vertex to the other. Let the edge groups be .
Now we can start to 'unfold' the graph： Let be a 2-complex associated to a set of generators and relations for and be 2-complexes associated to . The inclusion map induces cellular maps . Hence we have
Let be the mapping cylinder of . i.e. has boundary components and .
Let be the complex obtained by gluing together two copies of along , take it’s universal cover . Now the fundemental group of is, in some sense, the group obtained by doubling along each . i.e. is the fundamental group of the
graph graph of groups .
Now by studying the 1-skeleton of the complex , one is able to conclude that is Gromov hyperbolic with boundary, as expected.
Hence from groups with Sierpinski carpet boundary we are able to produce groups with sphere boundary. Now if Cannon’s conjecture is true, is fundamental group of some hyperbolic 3-manifold, together with Gabai’s result that are fundamental groups of hyperbolic surfaces, we would have that is the fundamental group of a hyperbolic 3-manifold with totally geodesic boundary components.
Well, since now we don’t have Cannon’s conjecture, there is still something we can conclude:
Definition: A n-dimensional Poincare duality group is a group which has group cohomology satisfying n-dimensional Poincare duality.
Those should be thought of as fundamental groups of manifolds in the level of homology. Well, I know nothing about group cohomologies, luckily we have:
is a n-dimensional Poincare duality group iff it’s torsion free and has integral Cech cohomology of
Great! In our case IS the sphere! So it’s a 3-dimensional Poincare duality group~ Now we have a splitting of over a bunch of 2-dimensional Poincare duality groups (namely ) it follows that is a Poincare duality pair.
It is not known whether all such pairs can be realized as fundamental groups of 3-manifolds with boundary. If so, then by Thurston’s geometrization we can also obtain what we derived assuming Cannon’s conjecture.