Projections in the curve complex arising from covering maps

Abstract

Let $P:\Sigma \to S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $\Pi :\mathcal{C}(S)\to \mathcal{C}(\Sigma )$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}(\Sigma )$ using minimal intersection number conditions and prove that it approximates a nearest point projection to $\Pi (\mathcal{C}(S))$. We also approximate hulls of finite sets of vertices in the curve complex, together with their corresponding nearest point projections, using intersection numbers.

27 pages

References in corpus (2)

• Uniform hyperbolicity of the curve graph via surgery sequences
• Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs