Cubulating Gromov-Thurston manifolds
1. Gromov-Thurston manifolds
These are compact negatively curved manifolds which are not hyperbolic. They arise as branched coverings of simple type arithmetic real hyperbolic manifolds.
Theorem 1 Gromov-Thurston groups are virtually special cubical, i.e. act properly and cocompactly on special cube complexes.
Fundamental groups of special cube complexes have nice properties:
- Inject into .
- Lots of separable subgroups.
A theorem of Agol asserts that hyperbolic groups which are cubical are virtually special cubical. I do not use this theorem, I provide instead a direct construction.
2. Bergeron-Haglund-Wise theory
A cube complex is a CW-complex with a metric which turns each cell into a Euclidean cube. If the link of vertices are flag complexes, the metric is locally . Locally + simply connected gives. In such a complex, there are natural hyperplanes, giving rise to a wall-space structure.
Example. According to Sageev, a collection of codimension 1 totally geodesic submanifolds in a hyperbolic manifold allows to construct a cube complex structure.
Simple type real arithmetic hyperbolic manifolds have enough arithmetic submanifolds to perform Sageev’s construction.
3. Sketch of proof
Start with a compact hyperbolic manifold admitting two codimension 1 totally geodesic submanifolds and , each separating , intersecting orthogonally. This splits into 4 convex pieces. Inserting more pieces produces a new manifold with a branched covering map to .
Theorem 2 From Sageev’s special cubulation of , we get a special cubulation of .
Merely need to adjust the cubulation of to the splitting defined by and . Apply Sageev’s construction to the whole collection of submanifolds. Use barycentric cubic subdivision.