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.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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