Geometrical compactifications of geodesic flows and path structures

Abstract

In this paper, we construct a geometrical compactification of the geodesic flow of non-compact complete hyperbolic surfaces $\Sigma$ without cusps having finitely generated fundamental group. We study the dynamical properties of the compactified flow, for which we show the existence of attractive circles at infinity. The geometric structure of ${\mathrm{T}}^1\Sigma$ for which this compactification is realized is the pair of one-dimensional distributions tangent to the stable and unstable horocyles of ${\mathrm{T}}^1\Sigma$. This is a Kleinian path structure, that is a quotient of an open subset of the flag space by a discrete subgroup $\Gamma$ of ${\mathrm{PGL}}_3(\mathbb{R})$. Our study relies on a detailed description of the dynamics of ${\mathrm{PGL}}_3(\mathbb{R})$ on the flag space, and on the construction of an explicit fundamental domain for the action of $\Gamma$ on its maximal open subset of discontinuity in the flag space.