Hierarchies and the curve complex,
Term I, 2010-2011


Week Date of Tuesday Tuesday Thursday Lecture notes Comments/Mistakes/Questions
2 Oct. 12 Introduction. Surfaces and curves. Intersection number. Simplical complexes. Tuesday Thursday See Thursday lecture for correct definition of inessential and peripheral. (The annulus, torus, Möbius band, and Klein bottle all give special cases.)
3 Oct. 19 Arc and curve complexes. Distance. The Farey graph. Annuli. Subsurface projection. Tuesday Thursday The Farey graph has a unique three-coloring.
4 Oct. 26 Disjoint, nested, overlapping subsurfaces. Behrstock's Lemma. Markings. Tuesday Thursday Oriented edges of the Farey graph are markings of S1,1. What are the twist and flip moves?
5 Nov. 2 The marking graph. Cocompactness. Finite stabilizers. Elementary moves. Local finiteness. Connectedness. The marking graph is quasi-isometric to MCG(S) equipped with the word metric. Tuesday Thursday A version of the Švarc-Milnor Lemma: Suppose that G is a group acting via isometries on a locally finite, connected graph X with finite vertex stabilizers and finite quotient. Then G is finitely generated and the orbit map from G to X is a quasi-isometry of the word metric with the path metric.
6 Nov. 9 Projection bounds. Distance estimates. Bounded geodesic image theorem. Tight geodesics, definition and existence. Tuesday Thursday Chris Leininger has given a proof of the Bounded Geodesic Image Theorem following Bowditch.
7 Nov. 16 Footprints. Subordinacy. Hierarchies. Hierarchies exist. Sigma. Structure of Sigma. Tuesday Thursday Do infinite geodesics exist in C(S)?
Thursday's lecture: M_k(n) should count unutilized domains only.
8 Nov. 23 More Sigma. Time order. Point order. "Finish" structure of Sigma. Tuesday Thursday
9 Nov. 30 Discuss applications. Slices and slice order. Transition slices and elementary moves. Resolutions. Tuesday Thursday In Thursday's lecture I need one more condition on τ0: the bottom pair is (gH, v0).
10 Dec. 7 Forward/backward paths. Sigma projection. Large link. Finish proof of the distance estimate. Applications. Tuesday Thursday