## 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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