Notes of Marc Burger’s third Oxford lecture 23-03-2017

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

1. To infinity (and beyond)

1.1. Compactification

We intend to describe the boundary of the representation variety ${Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))}$.

Theorem 1 (Burger-Pozzetti) If ${\rho:\Gamma\rightarrow Sp(2n,{\mathbb R})}$ is a maximal representation, then

1. Nontrivial elements do not have eigenvalues of modulus 1. In particular, translation lengths are ${>0}$.
2. If ${\gamma}$ and ${\eta}$ represent intersecting closed geodesics in ${\Sigma}$, then the following analogue of Collar Lemma holds,

$\displaystyle \begin{array}{rcl} (e^{\frac{\ell(\rho(\gamma))}{\sqrt{n}}}-1)(e^{\frac{\ell(\rho(\eta))}{\sqrt{n}}}-1)\geq 1. \end{array}$

In particular, the vector-valued length function ${\nu_\rho}$ never vanishes.

Corollary 2 (Parreau) The projectived length map

$\displaystyle \begin{array}{rcl} \mathbb{P}\circ\nu:Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))\rightarrow \mathbb{P}({\mathfrak{a}^+}^\Gamma) \end{array}$

has relatively compact image and any boundary point is the vector-valued length function of a ${\Gamma}$-action on a building associated to ${Sp(2n,{}^{\omega}{\mathbb R}_\lambda)}$ where ${{}^{\omega}{\mathbb R}_\lambda}$ is a Robinson field.

There is a loss of information.

Question 1. Does this building have special geometric properties?

Question 2. Is there a way to organize all these actions on buildings into a coherent compactification of ${Rep_{max}(\Gamma,Sp(2n,{\mathbb R}))}$?

The following is known since the 1980’s.

Theorem 3 (Skora) If ${n=1}$, boundary points are exactly length functions of actions on ${{\mathbb R}}$-trees with small stabilizers, i.e. stabilizers of germs of segments are either trivial or cyclic.

For ${n\geq 2}$, we need study sequences ${(\rho_k)}$ in ${Hom_{max}(\Gamma,Sp(2n,{\mathbb R}))}$ and the resulting actions an asymptotic cones of ${\mathcal{X}_n}$.

A sequence leaves to infinity if some marked point, say ${o=iId\in\mathcal{X}_n}$, is moved farther and farther away by some element of a fixed generating system ${S}$. Pick a sequence ${\lambda=(\lambda_k)_{k\geq 1}}$ such that

$\displaystyle \begin{array}{rcl} \max_{s\in S}d(\rho_k(s)o,o)=O(\lambda_k). \end{array}$

Pick a non-principal ultrafilter ${\omega}$. Form the corresponding asumptotic cone ${{}^\omega\mathcal{X}_\lambda}$. It is a complete ${CAT(0)}$ metric space, with an isometric action ${{}^\omega\rho_\lambda}$ of ${\Gamma}$.

Example. ${n=1}$. Assume ${\rho_k}$ pinches some closed geodesic ${\gamma}$, and does not affect curves disjoint from ${\gamma}$. Due to the Collar Lemma, any closed curve intersecting ${\gamma}$ has length tending to infinity at speed ${\lambda_k=\log(1/}$length${\rho_k(\gamma))}$. In the limit, ${{}^\omega\rho_\lambda(\gamma)}$ has a fixed point.

Definition 4 A simple closed geodesic ${c}$ on ${\Sigma}$ is special if

1. ${\ell({}^\omega\rho_\lambda(\gamma))=0}$ whenever ${\gamma}$ represents ${c}$.
2. For any closed geodesic ${c'}$ intersecting ${c}$, ${\ell({}^\omega\rho_\lambda(\gamma))>0}$ whenever ${\eta}$ represents ${c'}$.

There are at most ${3g-3}$ special geodesics.

Theorem 5 The isometric action ${{}^\omega\rho_\lambda}$ is faithful. Let ${\Sigma_v}$ be a component of the complement in ${\Sigma}$ of special geodesics. There is a dichotomy:

1. Either (PT): every curve in ${\Sigma_v}$ which is not a boundary component has ${\ell({}^\omega\rho_\lambda(c))>0}$.
2. Or (FP): ${\pi_1(\Sigma_v)}$ fixes a point in ${{}^\omega\mathcal{X}_\lambda}$.

This is used in the proof of

Theorem 6 (Burger-Pozzetti-Iozzi-Parreau) The ${\Gamma}$ action ${{}^\omega\rho_\lambda}$ on ${{}^\omega\mathcal{X}_\lambda}$ is small.

An other tool is the theory of maximal representations of surfaces with boundary. Indeed, the scale ${\lambda}$ is too large for certain components, thus a scale needs be chosen for each component. Only finitely many scales arise.

Personnally, I am not too fluent with buildings, I prefer the language of Robinson fields.