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Category Archives: Surfaces
Agol’s Virtual Haken Theorem (part 1)
I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3manifold topology (hat tip to Henry … Continue reading
The HallWitt identity
The purpose of this blog post is to try to give some insight into the “meaning” of the HallWitt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which … Continue reading
Posted in Groups, Lie groups, Surfaces, Visualization
Tagged commutators, gropes, HallWitt identity, visualization
1 Comment
Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet
In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the GaussBonnet theorem. 1. Triangles in Hyperbolic Space Suppose we are given … Continue reading
Polygonal words
Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sanghyun Kim on polygonal words in free groups. Their work is motivated by the following wellknown question of Gromov: Question(Gromov): Let be a oneended wordhyperbolic group. … Continue reading
Posted in Groups, Surfaces
Tagged double of free group, ends, Henry Wilton, hyperbolic groups, roundoff trick, Sanghyun Kim, scl, Stallings theorem on ends, surface subgroup
6 Comments
Minimal laminations with leaves of different conformal types
The “header image” for this blog is an example of an interesting construction in 2dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it … Continue reading
Bridgeman’s orthospectrum identity
Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The dimensional case is (in my opinion) the most beautiful, … Continue reading
SchwarzChristoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface
Hermann Amandus Schwarz (18431921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the … Continue reading