Curve complexes are rigid

TL;DR

This paper proves that any quasi-isometry of the curve complex closely resembles a simplicial automorphism, establishing the complex's rigidity and linking its geometric structure to the surface's topology.

Contribution

It demonstrates the rigidity of the curve complex by showing all quasi-isometries are close to automorphisms, revealing the complex's geometric and topological invariance.

Findings

Quasi-isometries are close to automorphisms of the curve complex

The quasi-isometry type determines the surface's homeomorphism class

Curve complex rigidity links geometry to surface topology

Abstract

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.