Curve complex

In mathematics, the curve complex is a simplicial complex C(S) associated to a finite type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups.

Let ${\displaystyle S}$ be a finite type connected oriented surface. More specifically, let ${\displaystyle S=S_{g,b,n}}$ be a connected oriented surface of genus ${\displaystyle g\geq 0}$ with ${\displaystyle b\geq 0}$ boundary components and ${\displaystyle n\geq 0}$ punctures.

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