** Convex cocompactness in real projective geometry **

Joint with Jeff Danciger and Francois Gueritaud.

Convex cocompactness makes sense in every space with a notion on convexity. The classical setting is hyperbolic geometry, where convex cocompact discrete groups are plentiful: uniform lattices, Schottky groups, quasi-Fuchsian groups.

In higher rank symmetric spaces, Kleiner-Leeb, Quint have shown that convex cocompactness implies cocompactness. Therefore one must switch to more flexible notions.

**1. Projective convex-cocompactness **

Definition 1Say that a discrete subgroup is -cc if it admits a convex cocompact action on some proper open convex subset of . If is strictly convex, one says that is strongly -cc.

We shall see that there are many examples, with good properties, and an interesting link to Anosov representations.

Note that if a discrete subgroup preserves a proper open convex subset in projective space, then it is automatically properly discontinuous, since it preserves Hilbert’s metric. Therefore the proper setting is that of Yves Benoist’s divisible convex sets.

**2. Examples **

** 2.1. Deformations **

Strong projective convex cocompactness is stable under small deformations. Hence, starting with classical convex cocompact subgroups of , one gets a lot of small deformations.

For cocompact lattices of , it turns out that all continuous deformations in are -cc (Koszul, Choi-Goldman, Benoist).

Anti de sitter quasi-Fuchsian groups. Uniform lattices of can be continuously deformed in . Mess has shown that all continuous deformations are -cc. We generalize this below.

** 2.2. -cc groups **

These are discrete subgroups of whose convex sets are contained in the set

** 2.3. Right-angled reflection groups **

Let be the Coxeter group on generators with matrix . It is a reflection group in . with walls defined by linear forms , .

Vinberg: is discrete in iff for every which do not commute,

Theorem 2is strongly -cc iff is Gromov hyperbolic and Vinberg’s inequalities are strict.

When it turns out to be symmetric and nondegenerate, the bilinear form such that has signature , is -cc.

**3. Link with Anosov representations **

Definition 3 (Labourie)Let be a hyperbolic group. A representation is Anosov if there exists a continuous, transverse and dynamics-preserving equivariant map of to the space of partial flags , where point projective hyperplane.

Labourie showed that Hitchin representations (i.e. deformations of repr obtained from Fuschsian ones composed with the irreducible representation of ) of surface groups are Anosov.

Anosov representations are quasi-isometric embeddings, and stable under deformation.

Theorem 4Let be a discrete subgroup of preserving a strictly convex open set . Then is strongly -cc iff is Gromov hyperbolic and its morphism to is Anosov.

** 3.1. Consequences **

New examples of strongly -cc groups. For even , all Hitchin representations are strongly -cc. This fails for odd: such representations do not preserve any convex set.

New examples of Anosov representations. Any hyperbolic RACG provides one.

**4. -cc in general **

Non-strong -cc is not stable under small deformations. For instance, the convex hull of the accumulation points of an orbit depends on the orbit. Therefore, we modify the definition a bit, by requiring that the convex set is the convex hull of the accumulation points of all orbits.

Theorem 5For this new definition, -cc is stable under small deformations. and embedding in higher dimensions. Such a group is finitely generated and quasi-isometrically embedded in . is hyperbolic iff it is strongly -cc.