Universal lattices and Property $τ$

Details Universal lattices and Property $τ$ Universal lattices and Property $τ$

2005-02-06

arxiv.org/abs/math/0502112v2

Mathematics

Group Theory

16 pages

Scientific paper

10.1007/s00222-005-0498-0

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 $(\SL_2(R) \ltimes R^2, R^2)$.

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