Hyperbolicity of the curve complex,
Term II, 2010


Week Date of Thursday Thursday Friday Lecture notes Comments/Mistakes
2 Jan. 21 Surfaces and curves. Intersection number. Bigon Criterion. C(S). Farey graph. Edge metric. Hempel's lemma. Continued fractions. Thursday Friday Can Hempel's lemma be improved?

Friday's lecture: I drew -2/3 instead of 2/3! Oops!

3 Jan. 28 Train tracks. Transverse measures. Foliated tie neighborhoods. Carrying. Splittings. The basic observation. Nesting criterion. Dimension. Thursday Friday Fixed characterization of the Farey graph.
4 Feb. 4 Keane property. Dense leaves. No lecture due to technical difficulties. Thursday In class we gave a non-constructive proof that pairs (track, measure) exist having the Keane property. Give a constructive proof.

Thursday's lecture - Keane should also imply that w is positive on every branch of τ. (Otherwise μ could contain non-singular boundary leaves.)

5 Feb. 11 Finish dense leaves. C(S) has infinite diameter. Coarse geometry. Thursday Friday Measures decompose as a finite sum of atomic measures on curves and minimal measures. Can you use "volume growth" to prove that the Farey graph is not quasi-isometric to H2 or T3.
6 Feb. 18 Triangles. Hyperbolicity. Combings. Combing implies chords. Combinatorial area. Isoperimetric inequalities. Thursday Friday If X is a tree then δ = 0. The converse is essentially true (R-trees). If (X, d) is Gromov hyperbolic then so is the quasi-isometric space (X, d/n); note that δ scales as well. Gromov-Hausdorff convergence. Question: are the paths in a slim combing quasi-geodesics?
7 Feb. 25 Chords imply subquadratic. Subquadratic implies linear. Linear implies stable. Stable implies slim. Singular flat structures. Thursday Friday A singular flat surface. Warning: some of the vertices have cone angle 3π/2.

Thursday's lecture - The maximum of (x + a)^2 + ((1-x) + a)^2 occurs when x is as large (or as small) as possible.

8 Mar. 4 Examples of half-translation surfaces. Geodesics. Flat annuli. Length. Systole map. Balance time. Thursday Friday Squared surfaces. Singular flat metrics. (Half-)translation surfaces. Abelian/quadratic differentials. Rational billard tables. Suspensions of interval exchange transformations...
9 Mar. 11 Wide annuli. Separation lemma. Minimal spines and systole length. Finish separation. Annulus inequality. Wide curves. Thursday Friday The isoperimetic function for the plane is x2/4π. Puzzle: what is the shortest arc dividing the equilateral triangle into two pieces of equal area?
10 Mar. 18 Review. Set of systoles has bounded diameter. Weighted multicurves and their squared surfaces. W ⊂ L part II. The combing. Flat length is convex. Averaging trick. Combing triangles are slim. Thursday Friday Have a good break.