Quasi-Random profinite groups

2012/02/19 by Bardestani, Mohammad, Mallahi-Karai, Keivan
#20C33 #20F #20P05 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1202.4194

Abstract

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of SL_k(ℤ/(pⁿℤ)) and Sp_2k(ℤ/(pⁿℤ)). Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups SL_k(ℤ_p) and Sp_2k(ℤ_p). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

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