Notes of Marc Burger’s first Oxford lecture 21-03-2017

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

1. From geometric structures to representation

Let {\Gamma} be a finitely presented group, and {G<Gl(n,{\mathbb R})} be defined by polynomials. Then {Hom(\Gamma,G)} is a real algebraic set. The representation variety is {Hom(\Gamma,G)/G} (not quite Hausdorff, see below).

Geometric structures on manifolds with fundamental group {\Gamma}, modelled on {G} acting on some homogeneous space {Y}, give rise to representations, their holonomy representations.

For instance, hyperbolic structures on a surface give rise to representations in {PSl(2,{\mathbb R})}. It turns out that this defines an injection of Teichmuller space into {Hom(\Gamma,PSl(2,{\mathbb R}))/PSl(2,{\mathbb R})}. It falls into one single component.

2. Parreau-Thurston compactification of {Hom(\Gamma,G)}

It is pretty general, but, for this exposition, we shall focus on {G=Sp(2n,{\mathbb R})}.

2.1. Geometry of Siegel’s half-space

{Sp(2n,{\mathbb R})} acts on the symmetric space

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

sometimes called Siegel’s half-space. {\mathcal{X}_n} has an invariant Riemannian metric for which the geodesic symmetry s an isometry. This metric is {CAT(0)}.

Elements of {Sp(2n,{\mathbb R})} can be viewed as block matrices {\begin{pmatrix} A & B \\ C & D \end{pmatrix}} such that {A^\top D -C^\top B=I}, {A^\top C}, {B^\top D} are symmetric. The action on {\mathcal{X}_n} is {Z\mapsto (AZ+B)(CZ+D)^{-1}}.

There is an explicit formula for the distance, as length of a vector-valued distance. Under {Sp(2n,{\mathbb R})}, every pair of points {(Z_1,Z_2)} of {\mathcal{X}_n} is equivalent to a unique pair {(iI,iD)} where {D} is diagonal with non-increasing entries {\geq 1}. Then the vector-valued distance is defined by

\displaystyle  \begin{array}{rcl}  \delta(Z_1,Z_2)=(\log d_1,\ldots,\log d_n). \end{array}

It satisfies a form of triangle inequality. The metric is

\displaystyle  \begin{array}{rcl}  d(Z_1,Z_2)=\sqrt{(\log d_1)^2+\cdots+(\log d_n)^2}. \end{array}

The higher rank analogue of translation length is the (unique) vector {\nu(g)} with non-negative entries of smallest Euclidean norm in the closure of {\{\delta(gZ,Z)\,;\, Z\in \mathcal{X}_n\}}. It turns out that {\nu} is continuous on {G} (a fact that would fail on Euclidean space).

2.2. Topology of the representation variety

{G} orbits in {Hom(\Gamma,G)} need not be closed. The source of all troubles are unipotent elements. To get rid of them, we stick to reductive representations, i.e. representations which are direct sums of irreducible representations. These are precisely those representations whose orbit is closed. Furthermore, every orbit contains a unique reductive representation in its closure.

Theorem 1 {Rep(\Gamma,G):=Hom_{red}(\Gamma,G)/G} is homeomorphic to a subset of some vector space defined by finitely many polynomial inequalities. This is algorithmic.

2.3. A projective embedding

Let {\mathfrak{a}^+={\mathbb R}_+^n}. Map a reductive representation to the sequence of its translation vectors {\nu(g)\in \mathfrak{a}^+}. This provides us with a map

\displaystyle  \begin{array}{rcl}  \nu:Rep(\Gamma,G)\rightarrow {\mathfrak{a}^+}^{\Gamma}. \end{array}

Next we compose with projectivization

Theorem 2 (Parreau 2010) The image of {P(\nu)} has compact closure. Every point of the closure is given by {[\nu\circ\rho]} where {\rho} is an isometric action of {\Gamma} on an affine building without global fixed point.

We shall see next time how to relate these actions to homomorphisms into {Sp(2n,F)} where {F} is a real field.

Question. Do these actions come from actions on {{\mathbb R}^n}-trees?

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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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