Injections of the complex of separating curves into the Torelli complex

TL;DR

This paper proves that for most compact orientable surfaces, superinjective maps from the separating curves complex to the Torelli complex are essentially geometric, induced by the extended mapping class group, with applications to homomorphisms in the Johnson kernel.

Contribution

It establishes a rigidity result linking superinjective maps to the extended mapping class group for a broad class of surfaces.

Findings

Superinjective maps are induced by extended mapping class group elements.

Injective homomorphisms from the Johnson kernel to the Torelli group are geometric.

The results hold for all but finitely many surfaces.

Abstract

We show that for all but finitely many compact orientable surfaces, any superinjective map from the complex of separating curves into the Torelli complex is induced by an element of the extended mapping class group. As an application, we prove that any injective homomorphism from a finite index subgroup of the Johnson kernel into the Torelli group for such a surface is induced by an element of the extended mapping class group.