Projections and hulls in the curve graph

Abstract

The curve graph \(C(S)\) associated to a surface \(S\) is a graph whose vertices are simple closed curves on \(S\) and whose edges are spanned by pairs of disjoint curves. This provides a combinatorial means for understanding the coarse geometry of mapping class groups, Teichmueller space and hyperbolic 3-manifolds. After presenting some basic definitions, I will describe a coarse analogue of a "convex hull" for a finite set of vertices in \(C(S)\) using only intersection number information. I then show how these results can be used to give a combinatorial approximation for nearest point projection maps to subgraphs of the curve graph which arise naturally from surface covering maps.