## Notes of Anne Giralt’s lecture

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 ${CAT(0)}$ cube complexes.

Fundamental groups of special ${CAT(0)}$ cube complexes have nice properties:

1. Inject into ${Gl(n,{\mathbb Z})}$.
2. Bi-orderable.
3. 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 ${CAT(0)}$ 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 ${CAT(0)}$. Locally ${CAT(0)}$ + simply connected gives${CAT(0)}$. 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 ${CAT(0)}$ 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 ${V}$ admitting two codimension 1 totally geodesic submanifolds ${V_1}$ and ${V_2}$, each separating ${V}$, intersecting orthogonally. This splits ${V}$ into 4 convex pieces. Inserting more pieces produces a new manifold ${V'}$ with a branched covering map to ${V}$.

Theorem 2 From Sageev’s special cubulation of ${V}$, we get a special cubulation of ${V'}$.

Merely need to adjust the cubulation of ${V}$ to the splitting defined by ${V_1}$ and ${V_2}$. Apply Sageev’s construction to the whole collection of submanifolds. Use barycentric cubic subdivision.