Title:

Covering maps and hulls in the curve complex

This thesis studies the coarse geometry of the curve complex using intersection number techniques. We show how weighted intersection numbers can be studied using appropriate singular Euclidean surfaces. We then introduce a coarse analogue of the convex hull of a finite set of vertices in the curve complex, called the short curve hull, and provide intersection number conditions to find nearest point projections to such hulls. We also obtain an upper bound for distances in the curve complex using a greedy algorithm due to Hempel. Covering maps between surfaces also play a significant part in this thesis. We give a new proof of a theorem of Rafi and Schleimer which states that a covering map between surfaces induces a natural quasiisometric embedding between their corresponding curve complexes. Our proof employs a distance estimate via a suitable hyperbolic 3manifold which arises from work on the proof of the Ending Lamination Theorem. We then define an operation using a given covering map and intersection number conditions and show that it approximates a nearest point projection to the image of RafiSchleimer's map. We also prove that this operation approximates a circumcentre of the orbit of a vertex in the curve complex under the deck transformation group of a regular cover
