On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface

TL;DR

This paper proves that for most finite-type hyperbolic surfaces, any homeomorphism of their geodesic lamination space corresponds to a surface homeomorphism, revealing a deep connection between lamination topology and surface symmetries.

Contribution

It establishes that homeomorphisms of the lamination space are induced by surface homeomorphisms for a broad class of hyperbolic surfaces, except certain low-complexity cases.

Findings

Homeomorphisms of lamination space correspond to surface homeomorphisms.

The result holds for all finite-type surfaces except spheres with ≤4 punctures and tori with ≤2 punctures.

The Thurston topology on lamination space is key to this correspondence.

Abstract

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.