# Exhausting Curve Complexes by Finite Rigid Sets on Nonorientable Surfaces

## TL;DR

This paper proves that for certain nonorientable surfaces, the curve complex can be built up from an increasing sequence of finite rigid sets, advancing understanding of their geometric structure.

## Contribution

It extends previous results by establishing exhaustion of the curve complex with finite rigid sets for broader classes of nonorientable surfaces.

## Key findings

- Exhaustion of curve complexes by finite rigid sets for specific nonorientable surfaces.
- Improves previous results on superrigid sets to rigid sets.
- Applicable to surfaces with genus 3 or higher, or with enough boundary components.

## Abstract

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(N)$ be the curve complex of $N$. We prove that if $(g,n) = (3,0)$ or $g + n \geq 5$, then there is an exhaustion of $\mathcal{C}(N)$ by a sequence of finite rigid sets. This improves the author's result on exhaustion of $\mathcal{C}(N)$ by a sequence of finite superrigid sets.