Category Archives: Groups

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

Posted in 3-manifolds, Groups, Hyperbolic geometry, Polyhedra | Tagged , , , , | 1 Comment

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 Crisp-Sageev-Sapir in the context of their … Continue reading

Posted in Ergodic Theory, Groups, Surfaces | Tagged , , , , , , | 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 Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following: Theorem … Continue reading

Posted in 3-manifolds, Ergodic Theory, Groups, Hyperbolic geometry | Tagged , , , , | 8 Comments

Agol’s Virtual Haken Theorem (part 2): Agol-Groves-Manning 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 Agol-Groves-Manning, and generalizes some earlier work they did a … Continue reading

Posted in Groups | Tagged , , , , , , | 15 Comments

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 3-manifold topology (hat tip to Henry … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , | 4 Comments

The Hall-Witt identity

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt 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 , , , | 1 Comment

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 | 2 Comments