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Category Archives: 3-manifolds
Hyperbolic Geometry Notes #5 – Mostow Rigidity
1. Mostow Rigidity For hyperbolic surfaces, Moduli space is quite large and complicated. However, in three dimensions Moduli space is trivial: Theorem 1 If is a homotopy equivalence of closed hyperbolic manifolds with , then is homotopic to an isometry. … Continue reading
Posted in 3-manifolds, Groups, Hyperbolic geometry, Uncategorized
3 Comments
Knots with small rational genus
I recently uploaded a paper to the arXiv entitled Knots with small rational genus, joint with Cameron Gordon. The genesis of this paper was a couple of nice (and related) talks at Caltech by Matthew Hedden and Jake Rasmussen in … Continue reading
Quasimorphisms from knot invariants
Last week, Michael Brandenbursky from the Technion gave a talk at Caltech on an interesting connection between knot theory and quasimorphisms. Michael’s paper on this subject may be obtained from the arXiv. Recall that given a group , a quasimorphism … Continue reading
Posted in 3-manifolds, Groups
Tagged 4-ball genus, braids, Cochran-Orr-Teichner, knot concordance, quasimorphisms, ribbon, signature, slice
2 Comments
Second variation formula for minimal surfaces
If is a smooth function on a manifold , and is a critical point of , recall that the Hessian is the quadratic form on (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of at … Continue reading
Surface subgroups – more details from Jeremy Kahn
Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety. Editorial … Continue reading
Posted in 3-manifolds, Ergodic Theory, Surfaces
Tagged Kahn, Markovic, pair of pants, surface groups
4 Comments
Surface subgroups in hyperbolic 3-manifolds
I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic -manifold contains a closed -injective surface. Equivalently, contains a closed surface subgroup. Apparently, Jeremy … Continue reading
Posted in 3-manifolds, Ergodic Theory, Surfaces
Tagged Bowen, geodesic flow, Hall's marriage theorem, Kahn, LERF, Markovic, pair of pants, surface groups, Waldhausen
10 Comments
The topological Cauchy-Schwarz inequality
I recently made the final edits to my paper “Positivity of the universal pairing in 3 dimensions”, written jointly with Mike Freedman and Kevin Walker, to appear in Jour. AMS. This paper is inspired by questions that arise in the … Continue reading
Posted in 3-manifolds, TQFT
Tagged 3-manifolds, Besson-Courtois-Gallot, compression body, Dijkgraaf-Witten, Hamilton, hyperbolic manifolds, JSJ decomposition, Miles Simon, minimal surface, Perelman, Ricci flow, scalar curvature, TQFT, unitary, universal pairing, volume entropy, volume inequality
3 Comments
