6 F eb 2 00 5 Universal lattices and property τ

We prove that the universal lattices – the groups G = SLd(R) where R = Z[x1, . . . , xk], have property τ for d ≥ 3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ -constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K2 of some finite commutative rings and a relative property T of (SL2(R)⋉ R , R). Introduction The groups SLd(O), where O is a ring of integers is in a number field K, have many common properties – they all have Kazhdan property T, a positive solution of the congruence subgroup problem and super rigidity. In [16], Y. Shalom conjectured that many of these properties are inherited from the group SLd(Z[x]). He called the groups SLd(Z[x1, . . . , xk]) universal lattices, because they can be mapped onto many lattices in groups SLd(K) for different fields K. Almost nothing is known about the representation theory of these groups. The main result in this paper is Theorem 5, which says that the universal lattices have property τ , provided that d ≥ 3. However unlike the classical lattices SLd(O) these groups do not have congruence subgroup property and have infinite (even infinitely generated) congruence kernel. Let G be a topological group and consider the space G̃ of all equivalence classes of unitary representations of G on some Hilbert space H. This space has a naturally defined topology, called the Fell topology, as explained in [12] §1.1 or [11] Chapter 3 for example. Let 1G denote the trivial 1-dimensional representation of G and let G̃0 be the set of representations in G̃ which do not contain 1G as a subrepresentation (i.e., do not have invariant vectors). Definition 1 A group is G is said to have Kazhdan property T if 1G is isolated from G̃0 in the Fell topology of G̃. 2000 Mathematics Subject Classification: Primary 20F69; Secondary 13M05, 19C20, 20G05, 20G35, 20H05, 22E40, 22E55.