Laminations g\'eod\'esiques plates

TL;DR

This paper generalizes geodesic laminations from hyperbolic surfaces to locally CAT(0) spaces, classifies laminations on flat surfaces, and constructs complex laminations on finite metric graphs.

Contribution

It introduces a new class of geodesic laminations on CAT(0) spaces and provides a classification theorem for flat surfaces, expanding the understanding of laminations beyond hyperbolic geometry.

Findings

Classified geodesic laminations on flat surfaces.

Most finite metric graphs support complex laminations.

Constructed laminations with uncountably many leaves.

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.