A note on the rigidity of unmeasured lamination spaces

TL;DR

This paper investigates the symmetries of the unmeasured lamination space of surfaces, establishing conditions under which these symmetries correspond uniquely to surface homeomorphisms, with specific exceptions for certain low-complexity surfaces.

Contribution

It proves that all auto-homeomorphisms of the unmeasured lamination space are induced by extended mapping classes, except for a few low-complexity surface cases.

Findings

Auto-homeomorphisms correspond to extended mapping classes for most surfaces.

Exceptions occur for spheres with ≤4 punctures, tori with ≤2 punctures, and genus 2 closed surfaces.

The result characterizes the rigidity of the lamination space's automorphism group.

Abstract

We show that every auto-homeomorphism of the unmeasured lamination space of an orientable surface of finite type is induced by a unique extended mapping class unless the surface is a sphere with at most four punctures or a torus with at most two punctures or a closed surface of genus 2.