London Algebra Colloquium abstract
Kazhdan quotients of Golod–Shafarevich groups
Mikhail Ershov (Virginia), 11th June 2009
Abstract
Informally speaking, a finitely generated group G is said to be Golod–Shafarevich
(with respect to a prime p) if it has a presentation with a “small”
set of relators, where relators are counted with different weights depending on
how deep they lie in the Zassenhaus p-filtration. Golod–Shafarevich groups
are known to behave like (non-abelian) free groups in many ways: for instance,
every Golod–Shafarevich group G has an infinite torsion quotient,
and the pro-p completion of G contains a non-abelian free pro-p group.
In this talk I will extend the list of known “largeness” properties of
Golod–Shafarevich groups by showing that they always have an infinite quotient with
Kazhdan’s property (T). An important consequence of this result is a positive
answer to a well-known question on non-amenability of Golod–Shafarevich
groups.