Wednesday 10:40-12, 736 Evans

Organizer: Ian Agol

This semester, this seminar will focus on work of Dani Wise

and his collaborators on residual finiteness for hyperbolic

groups with a quasi-convex hierarchy. His results give a

proof of a conjecture of Baumslag that torsion one-relator groups

are residually finite. Another important consequence is that Haken

hyperbolic 3-manifolds are virtually fibered. We will spend a lot

of time discussing background material on one-relator groups,

hyperbolic groups, hierarchies, ends of groups and group pairs,

cube complexes, reflection groups and Artin groups,

subgroup separability, small cancellation theory,

and other geometric group theory concepts.

Correction 9/12: As Yi pointed out in lecture today, I mixed up the terms "Hopfian" and "non-Hopfian".

The sketch of proof that BS(2,3) is non-Hopfian was correct, but I incorrectly called it Hopfian.

Source Papers:

Check out Henry Wilton's blog, which has information about several topics we are covering

Mccool and Schupp, One relator groups and HNN extensions

Michah Sageev, Ends of group pairs and Non-Positively Curved Cube Complexes

This paper (based on Sageev's UCB thesis) gives an equivalence between an action of

a group G on a cube complex and a subgroup H<G such that the pair (G,H) has more than one end.

This generalizes an equivalence between splittings of groups and actions on trees.

Bergeron and Wise, A boundary criterion for cubulation

Haglund and Wise, Special cube complexes

This paper gives a criterion for a group to be a subgroup of a right-angled Artin group, with

consequences for linearity and subgroup separability.

Haglund, Finite index subgroups of graph products

This paper generalizes Scott's separability argument to word-quasiconvex subgroups of

abstract right-angled reflection groups.

Rita Gitik; Mahan Mitra; Eliyahu Rips; Michah Sageev, Widths of subgroups

This paper analyzes intersections of quasiconvex subgroups of a hyperbolic group.

Daniel Wise, The residual finiteness of negatively curved polygons of finite groups

This paper considers the case of square complexes, which has some simplifications over the case of

cube complexes.

Haglund and Wise, A combination theorem for special cube complexes

This paper is part of the inductive step for a group with a malnormal quasiconvex hierarchy.

Daniel Wise, The structure of groups with a quasiconvex hierarchy

This gives a sketchy summary of Dani's results

Stefan Friedl, Notes on a talk of Dani Wise

Notes by Friedl from the conference in Montreal in April.

Hsu and Wise, Cubulating graphs of free groups with cyclic edge groups

This paper shows how to obtain cubulations of amalgams. The generalization of this technique is

needed for the inductive step.

Agol, The virtual fibering conjecture and related questions, talk at the 2009 Georgia Topology conference