On a stratification of the space of (projective) measured laminations

TL;DR

This paper introduces a natural stratification of the space of projective measured laminations on hyperbolic surfaces and proves a rigidity result linking homeomorphisms preserving this stratification to the extended mapping class group.

Contribution

It defines a new stratification of the lamination space and establishes a rigidity theorem connecting stratification-preserving homeomorphisms to the extended mapping class group.

Findings

Stratification of the lamination space is natural and well-defined.

Homeomorphisms preserving the stratification correspond to the extended mapping class group.

Fills a gap in the proof of the rigidity of the action on unmeasured lamination space.

Abstract

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identified with the extended mapping class group of the corresponding surface. We use this approach to fill a gap in the proof of the rigidity of the action of the extended mapping class group on the unmeasured laminations space.