| 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 |
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.