Universal lattices and Property $\tau$

Abstract

We prove that the universal lattices -- the groups $G=\SL_d(R)$ where $R=\mathbb{Z}[x_1,...,x_k]$, have property $\tau$ for $d\ge 3$. 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 $(\SL_2(R) \ltimes R^2, R^2)$.

16 pages

Cited by in corpus (12)

• Finite Simple Groups as Expanders
• Property $(T)$ for noncommutative universal lattices
• Commutator width in Chevalley groups
• On linear representations of Chevalley groups over commutative rings
• On the congruence kernel of isotropic groups over rings
• Expander Graphs in Pure and Applied Mathematics
• Finite simple groups of Lie type as expanders
• A substitute for Kazhdan's property (T) for universal non-lattices
• The IA-congruence kernel of high rank free Metabelian groups
• Subspace Arrangements and Property T
• Abstract homomorphisms of algebraic groups and applications
• Non-expander Cayley graphs of simple groups