Laminations g\'eod\'esiques 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 infinity of the universal cover is endowed with a invariant total cyclic order. Then we study these new objects on surfaces endowed with flat 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 flat structure. We also show that every finite metric graph, except four, is the support of a geodesic lamination with uncountably many leaves none of whose is eventually periodic.