Universal lattices and unbounded rank expanders
By Martin Kassabov. 2005
55 citations recorded by the source corpus.
We study the representations of non-commutative universal lattices and use them to compute lower bounds of the τ-constant for the commutative universal lattices Gd,k=SLd(ℤ[x1,...,xk]), for d≥3 with respect to several generating sets.As an application we show that the Cayley graphs of the finite groups $\text{SL}_{3k}(\mathbb{F}_{p})$ can be made expanders with a suitable choice of generators. This provides the first example of expander families of groups of Lie type, where the rank is not bounded and provides counter examples to two conjectures of A. Lubotzky and B. Weiss.
Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering
View Paper: https://www.semanticscholar.org/paper/54df9ba1d476a075af197b18f0fda1ce803c9253
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In Artificial Intelligence
In Discrete Mathematics and Combinatorics
In Electrical and Electronic Engineering
Citations in the corpus
- Property (T) for noncommutative universal lattices (2008, methodology): “n the suitable sense (see subsection 5.4). Then, as with the other proof, we finish the argument by using relative property (T) for (EL2(R)⋉R2,R2) or rather certain generalization of it established in [Ka2]. As an immediate application of Theorem 1.1, we obtain a simple example of a profinite group G ”
- Expander graphs in pure and applied mathematics (2011, methodology): “rank". But it turned out that this conjecture is true and it is now fully proved and its proof required an ensemble of very dierent methods. The big breakthrough came with two works of Kassabov [K1] and [K2] ([K1] was very much in uenced by [KN] which in turn was modeled on [Sh3]). Rather than ”