## Notes of Marc Burger’s second Oxford lecture 22-03-2017

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

1. Maximality

1.1. Toledo invariants

On ${\mathcal{X}_n}$, ${Sp(2n,{\mathbb R})}$ preserves a Riemannian metric ${g}$ and a complex structure ${J}$, hence a non-degenerate differential 2-form ${\omega}$. It turns out that ${d\omega=0}$. Indeed, ${d\omega}$ should be invariant under the geodesic symmetry, and this is impossible for a 3-form.

Given ${\rho:\Sigma\rightarrow Sp(2n,{\mathbb R})}$, pick an equivariant map ${f:\tilde \Sigma\rightarrow \mathcal{X}_n}$, pull-back ${\omega}$, get a ${\Gamma}$-invariant form on ${\tilde\Sigma}$, which descends to ${\Sigma}$, abusively denoted by ${f^*\omega}$.

Definition 1 The Toledo invariant is

$\displaystyle \begin{array}{rcl} T(\rho)=\frac{1}{2\pi}\int_\Sigma f^*\omega. \end{array}$

Theorem 2 (Goldman 1980) If ${n=1}$,

1. ${|T(\rho)|\leq 2g-2}$.
2. Gauss-Bonnet: if ${\rho}$ is the holonomy representation of a hyperbolic structure, ${T(\rho)=2g-2}$.
3. Equality is attained iff ${\rho}$ is the holonomy representation of a hyperbolic structure.

1.2. Maximal representations

In general,

1. ${T(\rho)\in{\mathbb Z}}$.
2. ${|T(\rho)|\leq n(2g-2)}$.

Definition 3 Say ${\rho}$ is maximal if ${T(\rho)=n(2g-2)}$.

The set of maximal representations, ${Hom_{max}(\Gamma,G)}$, is a union of connected components of ${Hom(\Gamma,G)}$, hence defined by finitely many polynomial inequations.

It is non-empty. Here is a construction. Let ${\rho_h:\Gamma\rightarrow Sp(2,{\mathbb R})}$ be the holonomy of a hyperbolic structure. Let ${\pi:Sp(2,{\mathbb R})\rightarrow Sp(2n,{\mathbb R})}$ be a direct sum of irreducible symplectic representations. Let ${\chi:\Gamma\rightarrow\{\pm 1\}}$ be a character. Then

$\displaystyle \begin{array}{rcl} \rho=(\pi\circ\rho_h)\otimes \chi \end{array}$

is a maximal representation.

Question. Does one obtain in this way at least one element of each connected component of ${Hom_{max}(\Gamma,G)}$?

If ${n=1}$, yes (there are ${2^{2g}}$ components).

If ${n\geq 3}$, yes (there are ${3\cdot 2^{2g}}$ components).

If ${n=2}$, no. The total number of connected components is ${3\cdot 2^{2g}+(2g-4)}$, of which ${2g-3}$ are freak components.

Theorem 4 If ${\rho}$ is a maximal representation, then orbit maps are quasi-isometric embeddings.

2. Positivity

If ${n=1}$, 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 ${\mathcal{L}({\mathbb R}^{2n})}$ denote the set of Lagrangian subspaces, i.e. ${n}$ dimensional subspaces of ${{\mathbb R}^{2n}}$ on which the symplectic form restricts to 0.

Lemma 5 ${Sp(2n,{\mathbb R})}$ acts transitively on pairs of transverse Lagrangians. Let ${(L_1,L_2,L_3)}$ be a triple of pairwise transverse Lagrangians. Since ${{\mathbb R}^{2n}=L_1\oplus L_3}$, let ${L_2}$ be the graph of ${T:L_1\rightarrow L_3}$. Then ${T}$ is symmetric and non-degenerate. It has a signature, an integer between ${-n}$ and ${n}$, which is an invariant of triples called the Maslov invariant.

Say a triple of Lagrangian is maximal if its Maslov invariant is ${n}$.

Theorem 6 If ${\rho}$ s maximal, there is a continuous equivariant map of ${\partial\tilde\Sigma}$ to ${\mathcal{L}({\mathbb R}^{2n})}$ which sends positively oriented triples to maximal triples.

This follows from the configuration of attractive and repulsive fixed points of elements of ${\Gamma}$.

2.2. Question

Are such representations related to geometric structures on the surface ?

Yes and no. If ${n=2}$, the ${\Gamma}$ 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.