Golod-Shafarevich Groups with Property (T) and Kac-Moody Groups

LIE GROUPS, REPRESENTATIONS AND DISCRETE MATH
Topic:	Golod-Shafarevich Groups with Property (T) and Kac-Moody Groups
Speaker:	Mikhail Ershov
Affiliation:	IAS
Date:	Tuesday, March 21
Time/Room:	10:15am - 11:15am/S-101

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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<t_0<1 such that 1-|X|t_0+\sum_{i=1}^{\infty}r_i t_0^i<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.