Feb 2016

The February meeting of GGSE was held at Warwick on 26th February

Apologies for not posting this sooner, but life got in the way.

The lectures were:
A new cubulation theorem for hyperbolic groups
Daniel Groves (U. Illinois Chicago)
Agol proved that a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex is virtually special. Wise's Quasiconvex Hierarchy Theorem says that a hyperbolic group acting cocompactly on a tree with quasiconvex and virtually special edge and vertex stabilizers is virtually special. I will talk about a common generalization of these two theorems: a hyperbolic group acting cocompactly on a CAT(0) cube complex with quasiconvex and virtually special cell stabilizers is virtually special. This is joint work with Jason Manning.

Ping Pong in CAT(0) cube complexes
Aditi Kar (Southampton)
I will describe some recent results of mine with Michah Sageev. Let G be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex X without a fixed point at infinity. We show that for any finite collection of simultaneously inessential subgroups {H1,…,Hk} in G, there exists an element g
of infinite order in G such that, for all
i, the subgroup generated by Hi and g is the free product Hi∗⟨g⟩

Examples of groups to which this applies are the Burger-Moses simple groups that arise as lattices in products of trees. We build a boundary of strongly separated ultrafilters and utilize the action of the group on the boundary to play ping pong. I will introduce all the terminology, describe the boundary if time permits and conclude with a summary of equivalent conditions for the reduced
C*-algebra of the group to be simple.

Sphere boundaries of hyperbolic groups
Nir Lazarovich (ETH)
We show that the boundary of a one-ended simply connected at infinity hyperbolic group with enough codimension-1 surface subgroups is homeomorphic to a sphere. By works of Markovic and Kahn-Markovic our result gives a new characterization of groups which are virtually fundamental groups of hyperbolic 3-manifolds. Joint work with B. Beeker.