QUASI-RANDOM PROFINITE GROUPS
Glasgow Mathematical Journal
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 SL k ((/ޚp n ))ޚ and Sp 2k ((/ޚp n .))ޚ 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
... per bounds for the supremal measure of a product-free measurable subset of the profinite groups SL k ޚ( p ) and Sp 2k ޚ( p ). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree. 2010 Mathematics Subject Classification. 20P05, 20F, 20C33 Introduction. A subset A of a group G is called product-free, if the equation xy = z has no solution with x, y, z ∈ A. Babai and Sós  asked if every finite group G has a product-free subset of size at least c|G| for a universal constant c > 0. This question was answered negatively by Gowers in his paper on quasi-random groups  where he proved that for sufficiently large prime p, the group G = PSL 2 ކ( p ) has no product-free subset of size cn 8/9 , where n is the order of G. A feature of this group that plays an essential role in the proof is that the minimal degree of a non-trivial representation of G is O(p). This property of G, called quasi-randomness by Gowers, is due to Frobenius and has been generalized by Landazuri and Seitz  to other families of finite simple groups of Lie type. Apart from its intrinsic interest, this theorem has found several important applications. To name a few, Nikolov and Pyber  , used Gowers' theorem to obtain an improved version of a recent theorem of Helfgott  and Shalev  on product decompositions of finite simple groups. Gowers' method has also been used in studying the image of the word maps on finite simple groups [19, 18] . The focus of this paper will be quasi-randomness for compact groups and, more specifically, profinite groups. We will be interested in the family G((/ޚp n ))ޚ where G is either the special linear or symplectic group. Our goal is to establish a lower bound on the minimal degree of all non-trivial representations and also the minimal degree https://www.cambridge.org/core/terms. https://doi.