Universal lattices and unbounded rank expanders

Abstract

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal latticeswith respect to several generating sets.As an application of the above result we show that the Cayley graphs of the finite groups SL 3k (Fp) can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.

Universal lattices and property tau

“…In this paper we will work only with the groups SL d (R), where d > 2 and R is a finitely generated commutative ring. The arguments can be easily generalized to any high rank Chevalley group over R. It is also possible to extend some parts of the argument to Chevalley groups over noncommutative rings [9].…”

“…If the reader is interested in the values of the Kazhdan constants and generalization of Theorem 5 to non commutative rings, he is encourage to go over the sequel of this paper [9]. It is interesting to note that using a different generating set it is possible to improve the the τ -constant for…”

The algebraization of Kazhdan’s property (T)

“…This seemingly technical issue turned out to be of importance, as it triggered the passage to working with general finitely generated commutative rings, thereby releasing a part of the theory from the burden of an ambient locally compact group. Kassabov observed that the commutativity of the ring multiplication operation is not required in the proof, and consequently we have [58] The main tool in the proof of the result is the spectral theorem for representations of abelian groups. By taking the spectral measure on the Pontrjagin dual R 2 , corresponding to almost EL 2 (R)-invariant vectors, one gets a sequence of almost EL 2 (R)-invariant measures with respect to the dual action on R 2 .…”

Property (T) for Kac–Moody groups over rings

“…The fact that the pairs (G, H i ) have relative property (T ) follows from Kassabov's theorem [Ka1,Theorem 1.2].…”