Golod-Shafarevich groups with property (T) and Kac-Moody groups
by
Mikhail Ershov
Institute for Advanced Study

A finitely generated group is called a Golod-Shafarevich group if it has a presentation < X|R > with the following property: There exists a prime number p and a real number 0 < t₀ < 1 such that 1-|X|t₀+∑_i=1^∞r_i t₀ⁱ < 0 where r_i is the number of defining relators which have degree i with respect to the dimension p-series.

Golod-Shararevich groups are always infinite and moreover behave like free groups in many ways. On the other hand, it is not clear if a Golod-Shafarevich group must have 'a lot of' finite quotients. The following is a well-known question of this type:

Is it true that Golod-Shafarevich groups never have property (tau)?

By a recent work of Lackenby, an affirmative answer to this question would have implied Thurston's virtual positive Betti number conjecture for arithmetic hyperbolic 3-manifolds. In this talk I will show that the answer to the above question is negative in general. Explicit examples of Golod-Shafarevich groups with property (tau) (in fact, (T)) are given by lattices in certain Kac-Moody groups over finite fields.

Date received: March 17, 2006