Exhaustion of $\mathcal{C}(N)$ via rigid expansions

TL;DR

This paper demonstrates that for a non-orientable surface of genus at least 6, a finite subgraph can be used to determine the entire curve complex's automorphisms, extending the understanding of its rigid structure.

Contribution

It introduces a finite subgraph whose rigid expansions exhaust the curve complex, enabling the characterization of automorphisms induced by surface homeomorphisms.

Findings

Existence of a finite subgraph $rak{X}$ for non-orientable surfaces of genus ≥6.

Rigid expansions of $rak{X}$ exhaust the entire curve complex $rak{C}(N)$.

Automorphisms of $rak{C}(N)$ restricted to $rak{X}$ are induced by homeomorphisms.

Abstract

Let $N$ be a connected closed non-orientable surface of genus at least 6. In this work we prove that there exists a finite subgraph $\mathfrak{X}$ (Irmak's finite rigid set from ``Elmas Irmak. Exhausting curve complexes by finite rigid sets on nonorientable surfaces. J.Topol.Anal., 16(2):261--289, 2024'') such that any graph endomorphism $\varphi$ of $\mathcal{C}(N)$ whose restriction to $\mathfrak{X}$ is (locally) injective, $\varphi$ is induced by a homeomorphism of $N$. To prove this, we first prove that $\mathfrak{X}$ and its rigid expansions exhaust $\mathcal{C}(N)$.