Examples of hyperbolic groups that have boundary homeomorphic to $S^2$?

Question

I am working on understanding Cannon's Conjecture which is the following:

Suppose that $G$ is an infinite, finitely presented group whose Cayley graph is Gromov-hyperbolic and whose space at infinity is the 2-sphere $S^2$. Then $G$ is a Kleinian group.

I am just trying to gather examples of groups that have such a boundary.

Thanks!

Answer 1

You should be looking at fundamental groups of closed hyperbolic three-manifolds, which is the torsion free case of the Cannon conjecture. Note that hyperbolic three space, $\mathbb{H}^3$, has boundary $S^2$. By Švarc–Milnor the fundamental groups of closed hyperbolic three-manifolds are quasi-isometric to $\mathbb{H}^3$ therefore have the same boundary.

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Mapping torus construction

One way to produce such manifolds, and their fundamental groups, is through the mapping torus construction. Let $S$ be a closed surface of genus greater than one and $f:S \to S$ a homeomorphism. You can construct a three manifold $$M=\frac{S \times [0,1]}{(x,0)\sim(f(x),1)}.$$ This construction, by a theorem of Thurston, gives a hyperbolic manifold exactly when $f$, up to isotopy, is a psuedo-Anosov. Pseudo-Anosov is equivalent to any power of $f$ not fixing the isotopy class of essential simple closed curves on the surface. You can describe $\pi_1(M)$ explicitly by an HNN extension in a similar way you use the Seifert–van Kampen theorem theorem to describe amalgamated products. A Primer on Mapping Class Groups by Farb and Margalit explains some ways to construct pseudo-Anosov's.

Fairly recently the virtually fibered conjecture was proved. Amazingly, the theorem proves all closed hyperbolic three manifolds have a finite cover of the above form. An important part of the proof is a result by Kahn and Marcovic which shows that hyperbolic three-manifold groups have "lots of" quasi-isometrically embedded surface subgroups (a strengthening of the surface subgroup conjecture). Marcovic has put out a criterion based off this idea. Basically, under the conditions of the Cannon conjecture, if all such groups have "enough" quasi-isometrically embedded surface subgroups the Cannon conjecture is true. Note that there is a big open question asking if all one ended hyperbolic groups contain surface subgroups (or even the stronger question of quasi-isometrically embedded surface subgroups).

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Using computers (SnapPy)

You can also use programs which have been created to study three manifolds. The one you should definitely know about is SnapPea/SnapPy. Here is an example in SnapPy:

Do M = Manifold('m004'), to name the manifold m004 M, then type M.browse() which will give a bunch of information in a new window, including the fundamental group.
This manifold is not closed but you can do a Dehn filling. In SnapPy you can do this in the browse window or in the command line. For example one Dehn filling example is M.dehn_fill( (4,3) ). When you have a hyperbolic manifold M you can type G=M.fundamental_group() to get a presentation(or view it in the browse window). In the specific example above we get:

Generators:
   a,b
Relators:
   aBAbaabAB
   abbaBAbABabbbaBAbAB

If you want matrices from that presentation you can type G.SL2C( 'x' ) where x is a word in your presentation. This will get into floating point/approximation errors.

There are tutorials on SnapPy website. Here is one youtube video which explains a bit and also explains a cool way to construct manifolds in SnapPy by just drawing knots or links (then you can fill if you want closed).

There is a program Twister which can work with SnapPy and produce manifolds from the mapping torus perspective.

Just because it is cool and I want to show it off, here is an example using the drawing tool. Typing N = Manifold() opens up the drawing screen. Now draw a knot or a link. Under tools click send to SnapPy. Here is my example:

We will now fill the knot to get a closed manifold(most of the time hyperbolic). My arbitrary choice is N.dehn_fill( '(3,2)' ). You can use N.browse() or N.fundamental_group() to view the fundamental group and in this case we have:

Generators:
   a,b
Relators:
   aBAbabAbaBABaBABabAbabABaBABaBAbabAbabABaBABabAbabAbaBABaBAbabAbabABaBABaaBABaBAbabAbabABaBABabAbabAbaBABaBAbabAbabABaBAB
   aBABabAbabAbaBABaBAbabAbabABaBABabaBABaBAbabAbabABaBABabAbabAbaBABaBAbbbABaBABabAbabAbaBABaBAbabAbabABaBABabaBABaBAbabAbabABaBABabAbabAbaBABaBAbbAB

Good luck getting any understanding of the Cannon conjecture from that!