Hyperbolicity of the graph of nonseparating multicurves

Abstract

A nonseparating multicurve on a surface $S$ of genus $g\ge 2$ with $m\ge 0$ punctures is a multicurve $c$ so that $S-c$ is connected. For $k\ge 1$ define the graph $\mathcal{N}\mathcal{C}\left(S,k\right)$ of nonseparating $k$–multicurves to be the graph whose vertices are nonseparating multicurves with $k$ components and where two such multicurves are connected by an edge of length one if they can be realized disjointly and differ by a single component. We show that if $k<g∕2+1$, then $\mathcal{N}\mathcal{C}\left(S,k\right)$ is hyperbolic.