(Tel Aviv University)
"The algebraization of Kazhdan's property (T)"
A group is said to have Kazhdan's property (T) if every
isometric (not necessarily linear) action of it on a Hilbert
space fixes a point. Following a brief discussion of this
important property and some geometric approaches to it, we
shall concentrate on recent developments of algebraic
nature, including connections to K-theory, particularly
discussing the following recent result:
Theorem. Let R be any finitely generated commutative ring with unit, and let EL(n,R) < GL(n,R) be the subgroup generated by the elementary matrices over R. Then for all n > 1+ Krull dimension R, this group has property (T). In particular, SL_n(Z[x_1, ... ,x_m]) has property (T) for all n > m+2.