
Recent Posts
 Bing’s wild involution
 StiefelWhitney cycles as intersections
 Schläfli – for lush, voluminous polyhedra
 Slightly elevated Teichmuller theory
 Mr Spock complexes (after Aitchison)
 Roots, Schottky semigroups, and Bandt’s Conjecture
 Taut foliations and positive forms
 Explosions – now in glorious 2D!
 Dipoles and Pixie Dust
 Mapping class groups: the next generation
 Groups quasiisometric to planes
 Div, grad, curl and all this
 A tale of two arithmetic lattices
 3manifolds everywhere
 kleinian, a tool for visualizing Kleinian groups
 Kähler manifolds and groups, part 2
 Kähler manifolds and groups, part 1
 Liouville illiouminated
 Scharlemann on Schoenflies
 You can solve the cube – with commutators!
 Chiral subsurface projection, asymmetric metrics and quasimorphisms
 Random groups contain surface subgroups
 wireframe, a tool for drawing surfaces
 Cube complexes, Reidemeister 3, zonohedra and the missing 8th region
 Orthocentricity
Blogroll
 0xDE
 Area 777
 Bluefawnpinkmanga
 Combinatorics and more
 Deep street soul
 Evaluating EDiscovery
 floerhomology
 Gaddeswarup
 Geometric Group Theory
 Godel's lost letter and P=NP
 Images des mathematiques
 Jim Woodring
 Language Log
 Letters of note
 Low dimensional topology
 Math Overflow
 Math/Art Blog
 Mathematics under the microscope
 nCategory Cafe
 Noncommutative geometry
 Paul Krugman
 Persiflage
 Preposterous Universe
 Questionable content
 Quomodocumque
 Real Climate
 Scott McCloud
 Secret blogging seminar
 Sketches of topology
 Tanya Khovanova
 Terry Tao
 Tim Gowers
 Tony Phillips
Books
Software
Recent Comments
Categories
 3manifolds (21)
 4manifolds (2)
 Algebraic Geometry (2)
 Algebraic Topology (1)
 Biology (2)
 Commentary (4)
 Complex analysis (11)
 Convex geometry (2)
 Diophantine approximation (1)
 Dynamics (13)
 Ergodic Theory (8)
 Euclidean Geometry (8)
 Foliations (2)
 Geometric structures (6)
 Groups (31)
 Hyperbolic geometry (25)
 Knot theory (1)
 Lie groups (8)
 Number theory (2)
 Overview (2)
 Polyhedra (3)
 Probability (1)
 Projective geometry (2)
 Psychology (3)
 Riemannian geometry (1)
 Rigidity (2)
 Special functions (2)
 Surfaces (20)
 Symplectic geometry (3)
 TQFT (1)
 Uncategorized (6)
 Visualization (10)
Meta
Category Archives: Groups
Random groups contain surface subgroups
A few weeks ago, Ian Agol, Vlad Markovic, Ursula Hamenstadt and I organized a “hot topics” workshop at MSRI with the title Surface subgroups and cube complexes. The conference was pretty well attended, and (I believe) was a big success; … Continue reading
Cube complexes, Reidemeister 3, zonohedra and the missing 8th region
There is an old puzzle which starts by asking: what is the next number in the sequence 1,2,4,? We are supposed to recognize the start of the sequence and answer that the next number is surely 8, because the first … Continue reading
Surface subgroups of Sapir’s group
Let be the free group on two generators, and let be the endomorphism defined on generators by and . We define Sapir’s group to be the ascending HNN extension This group was studied by CrispSageevSapir in the context of their … Continue reading
Posted in Ergodic Theory, Groups, Surfaces
Tagged ffolded surface, fatgraph, HNN extension, hyperbolic group, Sapir's group, Stallings folding, surface subgroup
12 Comments
Agol’s Virtual Haken Theorem (part 3): return of the hierarchies
Ian gave his second and third talks this afternoon, completing his (quite detailed) sketch of the proof of the Virtual Haken Theorem. Recall that after work of KahnMarkovic, Wise, HaglundWise and BergeronWise, the proof reduces to showing the following: Theorem … Continue reading
Agol’s Virtual Haken Theorem (part 2): AgolGrovesManning strike back
Today Jason Manning gave a talk on a vital ingredient in the proof of Agol’s theorem, which is a result in geometric group theory. The theorem is a joint project of AgolGrovesManning, and generalizes some earlier work they did a … Continue reading
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