Notes of Fanny Kassel’s Cambridge lecture 10-05-2017

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 1 Say that a discrete subgroup ${\Gamma\subset PGl(n+1,{\mathbb R})}$ is ${{\mathbb R} P^n}$-cc if it admits a convex cocompact action on some proper open convex subset ${\Omega}$ of ${{\mathbb R} P^n}$. If ${\Omega}$ is strictly convex, one says that ${\Gamma}$ is strongly ${{\mathbb R} P^n}$-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 ${PO(n,1)}$, one gets a lot of small deformations.

For cocompact lattices of ${PO(n,1)}$, it turns out that all continuous deformations in ${PSl(n+1,{\mathbb R})}$ are ${{\mathbb R} P^n}$-cc (Koszul, Choi-Goldman, Benoist).

Anti de sitter quasi-Fuchsian groups. Uniform lattices of ${PO(2,1)}$ can be continuously deformed in ${PO(2,2)}$. Mess has shown that all continuous deformations are ${{\mathbb R} P^n}$-cc. We generalize this below.

2.2. ${H^{p,q}}$-cc groups

These are discrete subgroups of ${PO(p,q+1)}$ whose convex sets are contained in the set

$\displaystyle \begin{array}{rcl} H^{p,q}:=\{[x]\in {\mathbb R} P^{p+q}\,;\,\langle x,x \rangle_{p,q+1}<0\}. \end{array}$

2.3. Right-angled reflection groups

Let ${\Gamma}$ be the Coxeter group on ${n}$ generators ${\gamma_i}$ with matrix ${M}$. It is a reflection group in ${{\mathbb R}^{n+1}}$. with walls defined by linear forms ${\alpha_i}$, ${\alpha(v_i)=1}$.

Vinberg: ${\Gamma}$ is discrete in ${Gl(n+1,{\mathbb R})}$ iff for every ${i,j}$ which do not commute,

$\displaystyle \begin{array}{rcl} \alpha_i(v_j)<0 \textrm{ and }\alpha_i(v_j)\alpha_j(v_i)\geq 1. \end{array}$

Theorem 2 ${\Gamma}$ is strongly ${{\mathbb R} P^n}$-cc iff ${\Gamma}$ is Gromov hyperbolic and Vinberg’s inequalities are strict.

When it turns out to be symmetric and nondegenerate, the bilinear form ${B}$ such that ${B(v_i,v_j)=\alpha_i(v_j)}$ has signature ${(p,q+1)}$, ${\Gamma}$ is ${H^{p,q}}$-cc.

Definition 3 (Labourie) Let ${\Gamma}$ be a hyperbolic group. A representation ${\rho:\Gamma\rightarrow PGl(n+1,{\mathbb R})}$ is Anosov if there exists a continuous, transverse and dynamics-preserving equivariant map of ${\partial G}$ to the space of partial flags ${(p,H)}$, where point ${p\subset H}$ projective hyperplane.

Labourie showed that Hitchin representations (i.e. deformations of repr obtained from Fuschsian ones composed with the irreducible representation of ${Sl(2,{\mathbb R})}$) of surface groups are Anosov.

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

Theorem 4 Let ${\Gamma}$ be a discrete subgroup of ${PGl(n+1,{\mathbb R})}$ preserving a strictly convex open set ${\Omega}$. Then ${\Gamma}$ is strongly ${{\mathbb R} P^n}$-cc iff ${\Gamma}$ is Gromov hyperbolic and its morphism to ${PGl(n+1,{\mathbb R})}$ is Anosov.

3.1. Consequences

New examples of strongly ${{\mathbb R} P^n}$-cc groups. For even ${n}$, all Hitchin representations are strongly ${{\mathbb R} P^n}$-cc. This fails for ${n}$ odd: such representations do not preserve any convex set.

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

4. ${{\mathbb R} P^n}$-cc in general

Non-strong ${{\mathbb R} P^n}$-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 5 For this new definition, ${{\mathbb R} P^n}$-cc is stable under small deformations. and embedding in higher dimensions. Such a group ${\Gamma}$ is finitely generated and quasi-isometrically embedded in ${PGl(n+1,{\mathbb R})}$. ${\Gamma}$ is hyperbolic iff it is strongly ${{\mathbb R} P^n}$-cc.