Universal lattices and unbounded rank expanders

2005/02/11 by Martin Kassabov, Kassabov, Martin · 1 citation
Mathematics · #05E15 #19C20 #20C33 #20G35 #20H05 #22E40 #22E55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20F69 #Secondary 05C25 #math.CO #math.GR #msc:05C25 #msc:05E15 #msc:19C20 #msc:20C33 #msc:20F69 #msc:20G35 #msc:20H05 #msc:22E40 #msc:22E55

paper · pdf · doi:10.48550/arxiv.math/0502237

32 pages

arxiv created 2005/02/11 · openalex publication_date 2005/02/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices G_d,k= \SL_d(\Z[x₁,...,x_k]) with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups \SL_3k(\F_p) 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.

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