Notes of Alessandra Iozzi’s Cambridge lecture 08-05-2017

Compactification of higher Teichmuller spaces

{\Sigma} closed surface of genus {g}. View Teichmuller space as injective and discrete representations of fundamental group to {PSl(2,{\mathbb R})}, up to conjugation. Replacing {PSl(2,{\mathbb R})} by a larger Lie groups leads to higher Teichmuller theory. This was started by Hitchin 25 years ago.

The part I am interested in today is the compactification of Teichmuller space. This is understood yet only in two cases,

– groups of Hermitian type: a subset of representations, maximal representations, correspond to Teichmuller space; maximal means that a certain numerical invariant, subject to a universal inequality, achieves its maximal possible value;

– groups of real adjoint split type: the Hitchin component plays this role.

The goal is to study ultralimits of maximal (resp. Hitchin) representations. I will focus on the maximal case.

1. Thurston-Parreau compactification

Map a representation to the (projectivized) set of translation lengths of group elements. Translation lengths are measured in the action on the symmetric space {X} of the receiving group {G}.

1.1. Ultralimits

This can be alternatively described in terms of ultralimits. Fix a non-principal ultrafilter {\omega}. Fix a finite generating set {S} of {\pi_1(\Sigma)}. Consider a sequence {\rho_k} of representations where some origin {o} in the symmetric space is moved farther and farther away. Pick a sequence of scales {\lambda_k} such that the maximal displacement of {o} is {\Theta(\lambda_k)}. Then there is a limiting isometric action {^\omega \rho_\lambda} on the ultralimit {^\omega X_\lambda} of {(X,o,d^X/\lambda_k)}. It has translation lengths.

Theorem 1 (Burger-Pozzetti 2015)

  1. {^\omega \rho_\lambda} is injective.
  2. {\Sigma} splits into a finite disjoint union of subsurfaces {\Sigma_v} along the collection {\mathcal{C}} of closed geodesics {\gamma} whose translation length is 0 but such that the translation length of every closed geodesic {\eta} that intersects {\gamma} is non-zero.
  3. The restriction of {^\omega \rho_\lambda} to the fundamental group of {\Sigma_v} (with appropriate choice of base-point in {\Sigma_v})
    • either has a global fixed point,
    • or every non-peripheral element has non-zero translation length.

1.2. Robinson fields

We can describe {^\omega X_\lambda} when {G=Sp(2n,{\mathbb R})} by merely replacing the reals by a larger real closed field. Start with the ultraproduct {{\mathbb R}_\omega}. Fix the infinitesimal {\sigma\in{\mathbb R}_\omega} corresponding to the sequence {e^{-\lambda_k}}. Consider {\mathcal{O}=\{x\in{\mathbb R}_\omega\,;\,|x|\leq \sigma^{-c}} for some {c\in{\mathbb R}\}}, and {\mathcal{I}=\{x\in{\mathbb R}_\omega\,;\,|x|\leq \sigma^{-c}} for all {c\in{\mathbb R}\}}. Then the Robinson field is

\displaystyle  \begin{array}{rcl}  {\mathbb R}_{\omega,\sigma}=\mathcal{O}/\mathcal{I}. \end{array}

This is a non-Archimedean field, with an absolute value {\|\cdot\|}. Define a distance on positive diagonal matrices as the {\ell^1}-norm of logs of eigenvalues. This extends into a pseudo-distance on Siegel’s upper half-space.

Theorem 2 (Parreau) {^\omega X_\lambda} is isometric to the Siegel upper half space associated with {{\mathbb R}_{\omega,\sigma}} modded out by vanishing distances.

1.3. Boundary maps

Maximal representations admit continuous equivariant maps from {\partial\pi_1(\Sigma)} to the space of triples of Lagrangian subspaces. Using it, one can define a cross-ratio on {\partial\pi_1(\Sigma)}. This allows to express translation lengths of hyperbolic elements. Say a pair {(a,b)\in\partial\pi_1(\Sigma)} is somewhat short if for every {x,y,z,t\in \partial\pi_1(\Sigma)} such that {x,a,y,z,b,t} is positively oriented, {\log\|[x,y,z,t]\|=0}.

Theorem 3 Any somewhat short geodesic is simple.

This is a key tool in the following theorem.

Corollary 4 (Burger-Iozzi-Parreau-Pozzetti) Assume that all hyperbolic elements have positive translation length is the limiting action. Then

  1. either there exists a syrface filling minimal measured lamination and an equivariant isometric embedding of the corresponding {{\mathbb R}}-tree into {^\omega X_\lambda} (note that the action of {\pi_1(\Sigma)} on this tree has small stabilizers),
  2. or {\pi_1(\Sigma)} acts properly discontinuously on {^\omega X_\lambda} and the locally {CAT(0)} quotient has positive injectivity radius.

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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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