** 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 .
- Bi-orderable.
- 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.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri PoincarĂ©, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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