** Geometric structures, compactifications of representation varieties, and non archimedean geometry, II **

**1. Maximality **

** 1.1. Toledo invariants **

On , preserves a Riemannian metric and a complex structure , hence a non-degenerate differential 2-form . It turns out that . Indeed, should be invariant under the geodesic symmetry, and this is impossible for a 3-form.

Given , pick an equivariant map , pull-back , get a -invariant form on , which descends to , abusively denoted by .

Definition 1The Toledo invariant is

Theorem 2 (Goldman 1980)If ,

- .
- Gauss-Bonnet: if is the holonomy representation of a hyperbolic structure, .
- Equality is attained iff is the holonomy representation of a hyperbolic structure.

** 1.2. Maximal representations **

In general,

- .
- .

Definition 3Say is maximal if .

The set of maximal representations, , is a union of connected components of , hence defined by finitely many polynomial inequations.

It is non-empty. Here is a construction. Let be the holonomy of a hyperbolic structure. Let be a direct sum of irreducible symplectic representations. Let be a character. Then

is a maximal representation.

**Question**. Does one obtain in this way at least one element of each connected component of ?

If , yes (there are components).

If , yes (there are components).

If , no. The total number of connected components is , of which are freak components.

Theorem 4If is a maximal representation, then orbit maps are quasi-isometric embeddings.

**2. Positivity **

If , for a maximal representation, there is an equivariant map between boundaries, and this map is an order preserving homeomorphism. This has a generalization.

** 2.1. Lagrangian Grassmannian **

Let denote the set of Lagrangian subspaces, i.e. dimensional subspaces of on which the symplectic form restricts to 0.

Lemma 5acts transitively on pairs of transverse Lagrangians. Let be a triple of pairwise transverse Lagrangians. Since , let be the graph of . Then is symmetric and non-degenerate. It has a signature, an integer between and , which is an invariant of triples called theMaslov invariant.

Say a triple of Lagrangian is *maximal* if its Maslov invariant is .

Theorem 6If s maximal, there is a continuous equivariant map of to which sends positively oriented triples to maximal triples.

This follows from the configuration of attractive and repulsive fixed points of elements of .

** 2.2. Question **

Are such representations related to geometric structures on the surface ?

Yes and no. If , the action on projective 3-space is properly discontinuous and cocompact on a domain of discontinuity, yielding a projective structure on a 3-manifold diffeomorphic to the unit tangent bundle of the surface.