Exhausting pants graphs of punctured spheres by finite rigid sets

Abstract

Let $S_{0,n}$ be an $n$-punctured sphere. For $n\ge 4$, we construct a sequence $({\mathcal{X}}_i{)}_{i\in \mathbb{N}}$ of finite rigid sets in the pants graph $\mathcal{P}(S_{0,n})$ such that ${\mathcal{X}}_1\subset {\mathcal{X}}_2\subset ...\subset \mathcal{P}(S_{0,n})$ and ${\bigcup }_{i\ge 1}{\mathcal{X}}_i=\mathcal{P}(S_{0,n})$.

12 pages, 6 figures