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/(pnℤ)) and Sp2k(ℤ/(pnℤ)). 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(ℤp) and Sp2k(ℤp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.
