Projections in the curve complex arising from covering maps

Let P: ∑ → S be a finite degree covering map between surfaces. Rafi and Schleimer showed that there is an induced quasi-isometric embedding Π: C(S) → C(∑) between the associated curve complexes. We define an operation on curves in C(∑) using minimal intersection number conditions and prove that it approximates a nearest point projection to Π(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.