The Margulis Lemma and the Thick and Thin Decomposition for Convex Real Projective Surfaces

Abstract

Convex real projective surfaces are quotients of simply convex domains in the real projective planeRP2under the properly discontinuous and free action of a projective automorphism group. Such surfaces carry Hilbert metrics defined by logarithms of cross ratios. We prove an analogous proposition to the Margulis lemma in hyperbolic geometry holding for such a surfaceΣwith a Hilbert metric. This allows us to decomposeΣinto thick and thin components. We show a compactness result that given a certain collection of simple closed curvesα1, …, αnonΣ, the subset of the deformation space of convex structures P(Σ) corresponding to structures where the Hilbert lengths ofα1, …, αnare bounded above is compact.