Notes of Marc Burger’s fourth Leverhulme lecture 25-04-2017

Towards higher Teichmuller theory, IV

1. To infinity and beyond

Today, ${G=Sp(2n,{\mathbb R})}$. We consider maximal representations ${\rho:\Gamma_g\rightarrow G}$. They are quasi-isometric embeddings. The class is deformation invariant. If ${n=1}$, they coincide with holonomies of hyperbolic structures on ${S_g}$.

The goal is to understand diverging sequences of maximal representations, from different points of view.

2. A simplified version of Parreau-Thurston compactification

Simplified means that I gave up explaining the Weyl chamber valued length function.

${G}$ acts on Siegel upper half-space

$\displaystyle \begin{array}{rcl} \mathcal{X}_n=\{Z=X+iY\,;\,X,Y\in Sym_n({\mathbb R}),\,Y\gg 0\}. \end{array}$

The translation length of an element ${g\in G}$ is given by the formula

$\displaystyle \begin{array}{rcl} \sqrt{2\sum_{i=1}^n (\log|\lambda_i|)^2}, \end{array}$

where ${\lambda_i}$ are the eigenvalues of ${g}$. Note that the characteristic polynomial of a symplectic matrix is palindromic, whence the sum of ${n}$ terms only. Define ${L_\rho}$ as the map ${\Gamma_g\rightarrow {\mathbb R}_+}$, ${\gamma\mapsto \ell(\rho(\gamma))}$.

Corollary 1 (Parreau 2010) The map

$\displaystyle \begin{array}{rcl} Hom_{max}(\Gamma_g,G)/G \rightarrow \mathbb{P}({\mathbb R}_+^\Gamma),\quad [\rho]\rightarrow [L_\rho] \end{array}$

has relatively compact image ${\mathcal{L}(S_g,n)}$. Any ${f}$ belonging to its closure isthe length function of a ${\Gamma_g}$-isometric action on a building associated to ${Sp(2n,{\mathbb R}_{\omega,\sigma})}$, where ${{\mathbb R}_{\omega,\sigma}}$ is an appropriate Robinson field.

This is not the end of the story.

Question 1. Do the limiting actions on buildings have special geometric properties?

Question 2. Can one organize the actions on buildings into a compactification of ${Hom_{max}(\Gamma_g,G)/G}$?

For instance, if ${n=1}$, Skora’s theorem asserts that the closure is precsely the set of actions with small edge stabilizers.

Definition 2 An isometric action of ${\Gamma_g}$ on a ${CAT(0)}$ space has small stabilizers if the stabilizer of a germ of segment is trivial or cyclic.

2.1. Results on question 1

Fix a non-principal ultrafilter ${\omega}$ on ${{\mathbb N}}$. Let ${(\rho_k)}$ be a sequence of representations. Fix a base point ${o}$ in ${\mathcal{X}_n}$. Say a sequence of scales ${(\lambda_k)_{k\geq 1}}$ is adapted if for some finite generating set ${S\subset\Gamma_g}$,

$\displaystyle \begin{array}{rcl} \lim_\omega \frac{1}{\lambda_k}\max_{\gamma\in S} d(\rho_k(\gamma)o,o)<+\infty \end{array}$

Then one sets ${\sigma=(e^{-\lambda_k})_{k\in{\mathbb N}}\in {\mathbb R}_\omega}$. Let

$\displaystyle \begin{array}{rcl} ^\omega \mathcal{X}_\lambda=\mathrm{Cone}((\mathcal{X}_n,o,\frac{1}{\lambda_k}d)_{k\in {\mathbb N}}). \end{array}$

This is a ${CAT(0)}$ space, thus translation lengths are well defined. There is a limiting action ${^\omega \rho_\lambda}$ of ${\Gamma_g}$.

Question. How are the elements with vanishing translation lengths look like?

It is true that ${\ell(^\omega \rho_\lambda(\gamma))=\lim_\omega \frac{1}{\lambda_k}\ell(\rho_k(\gamma))}$.

Theorem 3 (Pozzetti 2015) Fix a hyperbolic metric on ${S_g}$ (for convenience). Say a closed geodesic ${\gamma\in\Gamma_g}$ is special if

1. ${\ell(^\omega \rho_\lambda(\gamma))=0}$.
2. For all ${\eta\in \Gamma_g}$ that cross ${\gamma}$, ${\ell(^\omega \rho_\lambda(\eta))>0}$.

Then special curves split ${S_g}$ into subsurfaces with geodesic boundary ${S_v}$.

Theorem 4 (Burger-Pozzetti 2015) The limiting representation ${^\omega \rho_\lambda}$ is faithfull. For every ${x_v\in S_v}$, the following dichotomy holds:

1. either (PT): For every non-peripheral ${\gamma\in\pi_1(S_v,x_v)}$, ${\ell(\rho(\gamma))=0}$.
2. or (FP): The restriction of ${^\omega \rho_\lambda}$ to ${\pi_1(S_v,x_v)}$ has a global fixed point.

To go further, one needs treat surfaces with boundary ${S_v}$ separately. Note that maximality makes sense. Adapted scales for each ${S_v}$ can be chosen. The process ends because ${S_g}$ has finite complexity.

Theorem 5 (Burger-Iozzi-Parreau-Pozzetti 2017) The limiting representation ${^\omega \rho_\lambda}$ has small stabilizers.

The proof depends on the answer to Question 2.

2.2. Answer to Question 2

We follow Brumfiel’s real spectrum compactification of real algebraic spaces. Points there are homomorphisms of the ring of functions to real closed fields.

Point stabilizers, if not cyclic, contain surface groups and the restriction of the representation is again maximal. They act on the link of the fixed point. We want to view this as an action on a spherical building, associated to a certain residue field. Brumfiel tells us that this action belongs to the compactification, hence is a limit of maximal representations.

2.3. Proof of Theorem 5

Maximality passes to the limit (maximality is defined in terms of existence of a transversality preserving boundary map between Grassmannian Lagrangians; this is a difficult transversality result).

Germs of arc stabilizers, if not cyclic, contain surface groups and the restriction of the representation is again maximal. Since maximal representations do not fall into parabolic subgroups, such a subgroup cannot fix a direction, contradiction.