Bijections of geodesic lamination space preserving left Hausdorff convergence

TL;DR

This paper introduces a new asymmetric distance on geodesic lamination space and proves a rigidity result showing that bijections preserving this distance correspond exactly to the extended mapping class group actions.

Contribution

It defines the left Hausdorff distance on geodesic lamination space and establishes an isomorphism between its automorphisms and the extended mapping class group.

Findings

The left Hausdorff distance is a valid asymmetric metric on lamination space.

Bijections preserving left Hausdorff convergence are precisely extended mapping class group elements.

The extended mapping class group acts rigidly on the lamination space under this distance.

Abstract

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.