Bijections of geodesic lamination space preserving left Hausdorff convergence

Ken'Ichi Ohshika (1) , Athanase Papadopoulos (2)

1 Department of Mathematics, Osaka University
2 IRMA - Institut de Recherche Mathématique Avancée

We introduce an asymmetric distance function, which we call the ``left Hausdorff distance function", on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

Mots clés: Hausdorff distance; geodesic lamination space; 57M99 Keywords: Hyperbolic structure; AMS classification: 37E30; mapping class group; asymmetric Hausdorff distance; geodesic lamination; Thurston theory; complete lamination; finite lamination; approximable lamination; rigidity

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]

Ken'Ichi Ohshika, Athanase Papadopoulos. Bijections of geodesic lamination space preserving left Hausdorff convergence. 2018. ⟨hal-01782032v2⟩