Volume 14, issue 3 (2014)

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Hyperbolicity of the graph of nonseparating multicurves

Ursula Hamenstädt

Algebraic & Geometric Topology 14 (2014) 1759–1778
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 $\mathsc{N}\mathsc{C}\left(S,k\right)$ of nonseparating $k\phantom{\rule{0.3em}{0ex}}$–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, then $\mathsc{N}\mathsc{C}\left(S,k\right)$ is hyperbolic.

Keywords
multicurve graph, hyperbolicity
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 20F65, 57M99