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}))}?

1.2. Answer to question 1

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.

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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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