Tag: Gromov boundary

Graph of groups in relation to 3-manifolds

Conan5 Comments

(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 M with boundary, if the inclusion map i: \pi_1(\partial(M)) \rightarrow \pi_1(M) is not injective, then there exists a simple non-trivial loop in \partial M bounding an embedded disc in M.

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 V, edge set E, to each vertex v we associate a group G_v and to each edge e (say connecting v_1, v_2) we also associate a group G_e, together with a pair of injective homomorphisms f_1: G_e \rightarrow G_{v_1}, f_2: G_e \rightarrow G_{v_2}.

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.)

Let’s look at an application in this paper of Kleiner and Kapovich which I also talked about in an earlier post. Continue from that pervious post, now we know that

Theorem: Any hyperbolic group (plus obvious conditions, namely torsion free and does not split over a finite cyclic group) with 1-dimensional boundary has \partial_\infty G homeomorphic to \mathbb{S}^1, the Sierpinski carpet or the Menger curve.

When \partial_\infty G = \mathbb{S}^1, my wonderful advisor Dave Gabai proved that G would act discrete and cocompactly on \mathbb{H}^2 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 \mathbb{H}^3, we have:

Cannon’s conjecture: If hyperbolic group G has boundary \mathbb{S}^2, then G acts discretely and cocompactly on \mathbb{H}^3 by isometries.

This conjecture was also mentioned another pervious post. Turns our we do not know much about groups with \mathbb{S}^2 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 \mathbb{S}^2. 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 G 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 H_1, \cdots, H_k each H_i < G.

Define a graph of groups \mathcal{G} with two vertices both labeled G, with k edges, all going from one vertex to the other. Let the edge groups be H_1, \cdots, H_k.

Now we can start to 'unfold' the graph： Let X_G be a 2-complex associated to a set of generators and relations for G and X_i be 2-complexes associated to H_i. The inclusion map induces cellular maps h_i: X_i \rightarrow X_G. Hence we have

\displaystyle h: \sqcup_{i=1}^n X_i \rightarrow X_G

Let X be the mapping cylinder of h. i.e. X has boundary components \sqcup_{i=1}^n X_i and X_G.

Let DX be the complex obtained by gluing together two copies of X along \sqcup_{i=1}^n X_i, take it’s universal cover \widetilde{DX}. Now the fundemental group \hat{G} of DX is, in some sense, the group obtained by doubling G along each H_i. i.e. \hat{G} is the fundamental group of the graph graph of groups \mathcal{G}.

Now by studying the 1-skeleton of the complex \widetilde{DX}, one is able to conclude that \hat{G} is Gromov hyperbolic with \mathbb{S}^2 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, \hat{G} is fundamental group of some hyperbolic 3-manifold, together with Gabai’s result that H_i are fundamental groups of hyperbolic surfaces, we would have that G is the fundamental group of a hyperbolic 3-manifold with n 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:

Theorem: (Bestvina-Mess)

\Gamma is a n-dimensional Poincare duality group iff it’s torsion free and \partial \Gamma has integral Cech cohomology of \mathbb{S}^{n-1}

Great! In our case \partial \hat{G} IS the sphere! So it’s a 3-dimensional Poincare duality group~ Now we have a splitting of \hat{G} over a bunch of 2-dimensional Poincare duality groups (namely H_i) it follows that (G; H_1, \cdots, H_n) 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.

A remark on a mini-course by Kleiner in Sullivan’s 70th birthday

Conan2 Comments

I spent the last week on Long Island for Dennis Sullivan’s birthday conference. The conference is hosted in the brand new Simons center where great food is served everyday in the cafe (I think life-wise it’s a wonderful choice for doing a post-doc).

Anyways, aside from getting to know this super-cool person named Dennis, the talks there were interesting~ There are many things I found so exciting and can’t help to not say a few words about, however due to my laziness, I can only select one item to give a little stupid remark on:

So Bruce Kleiner gave a 3-lecture mini-course on boundaries of Gromov hyperbolic spaces (see this related post on a piece of his pervious work in the subject)

Cannon’s conjecture: Any Gromov hyperbolic group with \partial_\infty G \approx \mathbb{S}^2 acts discretely and cocompactly by isometries on \mathbb{H}^3.

As we all know, in the theory of Gromov hyperbolic spaces, we have the basic theorem that says if a groups acts on a space discretely and cocompactly by isometries, then the group (equipped with any word metric on its Cayley graph) is quasi-isometric to the space it acts on.

Since I borrowed professor Sullivan as an excuse for writing this post, let’s also state a partial converse of this theorem (which is more in the line of Cannon’s conjecture):

Theorem: (Sullivan, Gromov, Cannon-Swenson)
For G finitely generated, if G is quasi-isometric to \mathbb{H}^n for some n \geq 3, then G acts on \mathbb{H}^n discretely cocompactly by isometries.

This essentially says that due to the strong symmetries and hyperbolicity of \mathbb{H}^n, in this case quasi-isometry is enough to guarantee an action. (Such thing is of course not true in general, for example any finite group is quasi-isometric to any compact metric space, there’s no way such action exists.) In some sense being quasi-isometric is a much stronger condition once the spaces has large growth at infinity.

In light of the above two theorems we know that Cannon’s conjecture is equivalent to saying that any hyperbolic group with boundary \mathbb{S}^2 is quasi-isometric to \mathbb{H}^3.

At first glance this seems striking since knowing only the topology of the boundary and the fact that it’s hyperbolic, we need to conclude what the whole group looks like geometrically. However, the pervious post on one dimensional boundaries perhaps gives us some hint on the boundary can’t be anything we want. In fact it’s rather rigid due to the large symmetries of our hyperbolic group structure.

Having Cannon’s conjecture as a Holy Grail, they developed tools that give raise to some very elegant and inspring proofs of the conjecture in various special cases. For example:

Definition: A metric space M, is said to be Alfors \alpha-regular where \alpha is its Hausdorff dimension, if there exists constant C s.t. for any ball B(p, R) with R \leq \mbox{Diam}(M), we have:

C^{-1}R^\alpha \leq \mu(B(p,R)) \leq C R^\alpha

This is saying it’s of Hausdorff dimension \alpha in a very strong sense. (i.e. the Hausdorff \alpha measure behaves exactly like the regular Eculidean measure everywhere and in all scales).

For two disjoint continua C_1, C_2 in M, let \Gamma(C_1, C_2) denote the set of rectifiable curves connecting C_1 to C_2. For any density function \rho: M \rightarrow \mathbb{R}^+, we define the \rho-distance between C_1, C_2 to be \displaystyle \mbox{dist}_\rho(C_1, C_2) = \inf_{\gamma \in \Gamma(C_1, C_2)} \int_\gamma \rho.

Definition: The \alpha-modulus between C_1, C_2 is

\mbox{Mod}_\alpha(C_1, C_2) = \inf \{ \int_M \rho^\alpha \ | \ \mbox{dist}_\rho(C_1, C_2) \geq 1 \},

OK…I know this is a lot of seemingly random definitions to digest, let’s pause a little bit: Given two continua in our favorite \mathbb{R}^n, new we are of course Hausdorff dimension n, what’s the n-modulus between them?

This is equivalent to asking for a density function for scaling the metric so that the total n-dimensional volume of \mathbb{R}^n is as small as possible but yet the length of any curve connecting C_1, \ C_2 is larger than 1.

So intuitively we want to put large density between the sets whenever they are close together. Since we are integrating the n-th power for volume (suppose n>1, since our set is path connected it’s dimension is at least 1), we would want the density as ‘spread out’ as possible while keeping the arc-length property. Hence one observation is this modulus depends on the pair of closest points and the diameter of the sets.

The relative distance between C_1, C_2 is \displaystyle \Delta (C_1, C_2) = \frac{\inf \{ d(p_1, p_2) \ | \ p_1 \in C_1, \ p_2 \in C_2 \} }{ \min \{ \mbox{Diam}(C_1), \mbox{Diam}(C_2) \} }

We say M is \alpha-Loewner if the \alpha modulus between any two continua is controlled above and below by their relative distance, i.e. there exists increasing functions \phi, \psi: [0, \infty) \rightarrow [0, \infty) s.t. for all C_1, C_2,

\phi(\Delta(C_1, C_2)) \leq \mbox{Mod}_\alpha(C_1, C_2) \leq \psi(\Delta(C_1, C_2))

Those spaces are, in some sense, regular with respect to it’s metric and measure.

Theorem: If \partial_\infty G is Alfors 2-regular and 2-Loewner, homeomorphic to \mathbb{S}^2, then G acts discrete cocompactly on \mathbb{H}^3 by isometries.

Most of the material appeared in the talk can be found in their paper.

There are many other talks I found very interesting, especially that of Kenneth Bromberg, Mario Bonk and Peter Jones. Unfortunately I had to miss Curt McMullen, Yair Minski and Shishikura…

Gromov boundary of hyperbolic groups

Conan6 Comments

As we have seen in pervious posts, the Cayley graphs of groups equipped with the word metric is a very special class of geodesic metric space – they are graphs that have tons of symmetries. Because of that symmetry, we can’t construct groups with any kind of Gromov boundary we want. In fact, there are only few possibilities and they look funny. In this post I want to introduce a result of Misha Kapovich and Bruce Kleiner that says:

Let G be a hyperbolic group that’s not a semidirect product H \ltimes N where N is finite or virtually cyclic. (In those cases the boundary of G can be obtained from the boundary of H$).

Theorem: When G has 1-dimensional boundary, then the boundary is homeomorphic to either a Sierpinski carpet, a Menger curve or S^1.

OK. So what are those spaces? (don’t worry, I had no clue about what a ‘Menger curve’ is before reading this paper).

The Sierpinski carpet

(I believe most people have seen this one)

Start with the unit square, divide it into nine equal smaller squares, delete the middle one.

Repeat the process to the eight remaining squares.

and repeat…

Of course we then take the infinite intersection to get a space with no interior.

Proposition: The Sierpinski carpet is (covering) 1-dimensional, connected, locally connected, has no local cut point (meaning we cannot make any open subset of it disconnected by removing a point).

Theorem: Any compact metrizable planar space satisfying the above property is a Sierpinski carpet.

The Menger curve

Now we go to \mathbb{R}^3, the Menger curve is the intersection of the Sierpinski carpet times the unit interval, one in each of the x, y, z direction.

Equivalently, we may take the unit cube [0,1]^3, subtract the following seven smaller cubes in the middle:

In the next stage, we delete the middle ‘cross’ from each of the remaining 20 cubes:

Proceed, take intersection.

Proposition: The Menger curve is 1-dimensional, connected, locally connected, has no local cut point.

Note this is one dimensional because we can decompose the ‘curve’ to pieces of arbitrary small diameter by cutting along thin rectangular tubes, meaning if we take those pieces and slightly thicken them there is no triple intersections.

Theorem: Any compact metrizable nowhere planar (meaning no open set of it can be embedded in the plane) space satisfying the above property is a Menger curve.

Now we look at our theorem, infact the proof is merely a translation from the conditions on the group to topological properties of the boundary and then seeing the boundary as a topological space satisfies our universal properties.

A group being Gromov hyperbolic implies the boundary is compact metrizable.

No splitting over finite or virtually cyclic group implies the boundary is connected, locally connected and if it’s not S^1, then it has no local cut point.

Now what remains is to show, for groups, if the boundary is not planar then it’s nowhere planar. This is an easy argument using the fact that the group acts minimally on the boundary.

Please refer to first part of their paper for details and full proof of the theorem.

Remark: When study classical Polish-school topology, I never understood how on earth would one need all those universal properties (i.e. any xxx space is a xxx, usually comes with a long condition include ten or so items >.<). Now I see in fact such thing can be powerful. i.e. sometimes this allows us to actually get a grib on what does some completely unimaginable spaces actually look like!

Another wonderful example of this is the recent work of S. Hensel and P. Przytycki and the even more recent work of David Gabai which shows ending lamination spaces are Nobeling curves.