Hyperbolicity of the graph of nonseparating multicurves

Abstract

A nonseparating multicurve on a surface S of genus g ≥ 2 with m ≥ 0 punctures is a multicurve c so that S - c is connected. For k ≥ 1 define the graph NC(S, k) 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