Geometry, Topology and Dynamics Seminar
Mikhail Ershov
University of Chicago
Golod-Shafarevich groups with Property (T) and Kac-Moody groups
Abstract: Informally speaking, a finitely generated group is called Golod-Shafarevich if
it has a presentation with a "small" set of relators. In 1964, Golod and
Shafarevich proved that groups satisfying such condition are necessarily
infinite and used this criterion to solve two outstanding problems: the
construction of infinite finitely generated periodic groups and the construction
of infinite Hilbert class field towers. An important class of Golod-Shafarevich
groups consists of the fundamental groups of compact hyperbolic 3-manifolds or,
equivalently, torsion-free lattices in SO(3,1). In 1983, Lubotzky used this fact
to prove that arithmetic lattices in SO(3,1) do not have the congruence subgroup
property. More recently, Lubotzky and Zelmanov proposed a group-theoretic
approach (based on Golod-Shafarevich techniques) to an even more ambitious
problem, Thurston's virtual positive Betti number conjecture. This approach led
to the following question: is it true that Golod-Shafarevich groups never have
property (\tau)? I will show that the answer to the above question is negative
in general and describe examples of Golod-Shafarevich groups with property
(\tau) (in fact, property (T)) which are given by lattices in certain
topological Kac-Moody groups over finite fields.
Monday April 16, 2007 at 3:00 PM in SEO 636