Universal lattices and property tau
Abstract
We prove that the universal lattices  the groups $G=\SL_d(R)$ where $R=\Z[x_1,...,x_k]$, have property $\tau$ for $d\geq 3$. This provides the first example of linear groups with $\tau$ which do not come from arithmetic groups. We also give a lower bound for the expanding 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 $K_2$ of some finite commutative rings and a relative property \emph{T} of $(\SL_2(R) \ltimes R^2, R^2)$.
 Publication:

Inventiones Mathematicae
 Pub Date:
 March 2006
 DOI:
 10.1007/s0022200504980
 arXiv:
 arXiv:math/0502112
 Bibcode:
 2006InMat.165..209K
 Keywords:

 Group Theory;
 Representation Theory;
 20F69;
 20G05;
 20H05;
 19C20
 EPrint:
 16 pages