# Exhausting Curve Complexes by Finite Superrigid Sets on Nonorientable Surfaces

## TL;DR

This paper proves that for most nonorientable surfaces, their curve complexes can be built up from an increasing sequence of finite superrigid sets, revealing structural insights into their topology.

## Contribution

It establishes the existence of an exhaustion of the curve complex by finite superrigid sets for a broad class of nonorientable surfaces, excluding specific low-complexity cases.

## Key findings

- Excludes the cases (g, n) = (1,2) and g + n = 4.
- Constructs an explicit sequence of finite superrigid sets.
- Shows the curve complex is exhausted by these finite sets.

## 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 $(g, n) \neq (1,2)$ and $g + n \neq 4$, then there is an exhaustion of $\mathcal{C}(N)$ by a sequence of finite superrigid sets.