Finite rigid sets and the separating curve complex

TL;DR

This paper constructs finite rigid subcomplexes within the separating curve complex for closed surfaces of genus at least 3, which are preserved under incidence-preserving maps, advancing understanding of the complex's structure.

Contribution

It introduces finite rigid subcomplexes of the separating curve complex for genus ≥ 3 surfaces, a novel step in understanding their automorphisms.

Findings

Finite rigid subcomplexes exist in the separating curve complex for genus ≥ 3 surfaces.

These subcomplexes are preserved under incidence-preserving maps.

The results contribute to the study of automorphisms of the separating curve complex.

Abstract

For each closed surface of genus $g\ge3$, we find a finite subcomplex of the separating curve complex that is rigid with respect to incidence-preserving maps.