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.

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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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