Exhaustion of the curve graph via rigid expansions

TL;DR

This paper demonstrates that a finite set of curves on a surface can generate the entire curve graph through iterated rigid expansions, linking to and extending previous finite rigid set constructions.

Contribution

It introduces a new finite set of curves whose rigid expansions exhaust the curve graph, connecting to and expanding upon Aramayona and Leiniger's finite rigid set.

Findings

Finite set of curves generates the curve graph via rigid expansions

Establishes a connection to Aramayona and Leiniger's finite rigid set

Provides a new method for understanding the structure of the curve graph

Abstract

For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.