Homeomorphisms and intersection number

TL;DR

This paper establishes rigidity results for automorphism groups of measured lamination spaces on hyperbolic surfaces, showing that certain homeomorphisms preserving intersection properties are highly constrained.

Contribution

It proves that homeomorphisms preserving intersection structures in measured lamination spaces are essentially geometric automorphisms, revealing deep rigidity properties.

Findings

Homeomorphisms of ML(S) preserving intersection are rigid

Homeomorphisms of PML(S) preserving zero sets are constrained

Results apply to closed hyperbolic surfaces

Abstract

We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorphisms of PML(S) that preserve the zero sets of these intersection functions.