[Paper Review] Anosov Flows, Surface Groups and Curves in Projective Space

François Labourie|ArXiv.org|Jan 19, 2004

Geometric and Algebraic Topology5 references4 citations

TL;DR

This paper establishes that representations in Hitchin's component of surface groups into PSL(n,R) correspond to holonomies of convex projective structures on surfaces, generalizing the classical Teichm"uller theory for n=2 and the convex RP^2 geometry for n=3. Using Higgs bundle techniques and geometric analysis, the authors prove that such representations are discrete, faithful, and preserve a Frenet curve in projective space, with the component homeomorphic to a ball of dimension χ(S)(1−n²).

ABSTRACT

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic)

Motivation & Objective

• To generalize the geometric interpretation of Hitchin's component from n=2 (Teichm"uller space) and n=3 (convex real projective structures) to higher n.
• To establish that representations in the Hitchin component preserve a Frenet curve in projective space, extending the notion of convex curves.
• To prove that such representations are discrete and faithful, using geometric and algebraic techniques beyond Higgs bundle methods.
• To analyze the action of outer automorphisms on the Hitchin component and characterize its topology and dynamics.

Proposed method

• Use of Higgs bundle techniques to analyze irreducibility and topology of the representation space.
• Definition and study of hyperconvex and Frenet curves in projective space P(R^n), requiring direct sums of osculating flags at distinct points.
• Construction of a continuous, ρ-equivariant map to the Grassmannian of p-planes, satisfying the Frenet flag condition.
• Application of the limit process on boundary points to derive invariance properties under group actions.
• Use of the non-existence of parallel sections in the endomorphism bundle to prove irreducibility of the representation.
• Employment of algebraic group theory and Lie algebra arguments to show that if a representation preserves a subspace under certain intersection conditions, it cannot be irreducible unless the group is reducible.

Experimental results

Research questions

• RQ1 Do representations in the Hitchin component of π₁(S) into PSL(n,R) preserve a Frenet curve in P(R^n) for n > 3?
• RQ2 Can the geometric structure of the Hitchin component be characterized as the holonomy of a convex projective structure on the surface?
• RQ3 Are all representations in the Hitchin component discrete and faithful, and how can this be shown beyond Higgs bundle techniques?
• RQ4 What is the role of the outer automorphism group of π₁(S) in acting on the Hitchin component?
• RQ5 Under what conditions does a representation preserving a subspace in the Grassmannian fail to be irreducible?

Key findings

• The Hitchin component of Rep(π₁(S), PSL(n,R)) is homeomorphic to a ball of dimension χ(S)(1−n²), extending Hitchin’s result.
• For n odd, there is one Hitchin component; for n even, there are two isomorphic components.
• Every representation in the Hitchin component preserves a Frenet curve in P(R^n), i.e., a continuous, hyperconvex curve with a well-defined osculating flag satisfying direct sum conditions.
• The existence of a ρ-equivariant Frenet curve implies that the representation is discrete and faithful.
• If a representation preserves a (n−k)-plane A such that dim(ξ^{k+1}(y) ∩ A) ≥ 1 on a non-empty open set of the boundary, then the restriction to a finite index subgroup is not irreducible.
• A subgroup of SL(n,R) with all elements real split and all finite index subgroups irreducible must be discrete, proving faithfulness under these conditions.