QUASI-RANDOM PROFINITE GROUPS

Inspired by Gowers' seminal paper (W. T. Gowers, Comb. Probab. Comput. 17(3) (2008), 363-387, we will investigate quasi-randomness for profinite groups. We will obtain bounds for the minimal degree of non-trivial representations of SLk(Z/(p(n)Z)) and Sp(2k)(Z/(p(n)Z)). Our method also delivers a lower bound for the minimal degree of a faithful representation of 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 SLk(Z(p)) and Sp(2k)(Z(p)). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.