Haken 3-manifold

Description

It is a 3-manifold such that we can find a ${\pi }_1$-injective embedded closed surface $\Sigma$ in $M$. This is analogous to the way the Klassifikation der Geschlossenen Flächen is done. Cut until we get discs. Here we want to cut until we get $3$-balls.

Definition

Properties

Theorem

For a Haken manifold ${\pi }_1(M)$ determines $M$.

Conjecture Theorem (Agol 13)

Are all hyperbolic 3-manifolds virtual Haken? Meaning, they is a finite cover which is Haken.

Theorem (Kahn-Kackovic)

Leet $M^3$ be a closed hyperbolic. There exists a closed immersed Quasi-convex surface ${\pi }_1$-inj. surface.

The proof uses a lot of Pair of pants. From this we get that the Fundamental group has a lot of quasi-convex surface subgroup.

Sageev and Bergerson-Wise show that $\Gamma$ acts properly and coccompactly on a CAT(0) Cube complex. CAT(0) of cube complex can be verified locally. A non-CAT(0)-complex might be one where three squares meet (but they are not the faces of a cube). At that point we have positive curvature. The main difference between a cube complex and a simplicial structure is that we can easily define hyperplanes.

There is some connection to Bass-Serre trees as well!

Conjecture (Wise) Theorem (Agol)

Let $\Gamma$ be a Hyperbolic group acting properly and cocompactly on a CAT(0) Cube complex. Then there exists a finite index subgroup which also embeds into a Right-angled Artin group. Meaning, $\Gamma$ is virtually Special group.

Definition

A Special group is a quasi-convexly in a RAAGs embedded group.

Examples

Not all $3$-manifolds are Haken

This is analogous to the group having some problems (they have torsion).