Laminations géodésiques plates

Abstract

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0) , complete, geodesic metric spaces, whose boundary at infi- nity of the universal cover is endowed with an invariant total cyclic order. Then we study these new objects on surfaces endowed with half-translation structures and on finite metric graphs. The main result of the paper is a theorem of classification of geodesic laminations on a compact surface endowed with a half-translation structure. We also show that every finite metric fat graph, outside four homeomorphism classes, is the support of a geodesic lamination with uncountably many leaves none of which is eventually periodic