Exhausting curve complexes by finite rigid sets

TL;DR

This paper proves that the curve complex of a finite-type surface can be exhausted by an increasing sequence of finite rigid sets, providing a new structural understanding of these complexes.

Contribution

It introduces a method to exhaust the curve complex with finite rigid sets, advancing the understanding of its combinatorial and geometric structure.

Findings

Curve complex can be exhausted by finite rigid sets

Provides a new approach to studying the structure of curve complexes

Enhances understanding of the rigidity properties of curve complexes

Abstract

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.