QUASI-RANDOM PROFINITE GROUPS

MOHAMMAD BARDESTANI, KEIVAN MALLAHI-KARAI
2014 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
more » ... 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 [1] 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 [6] 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 [13] 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 [15] , used Gowers' theorem to obtain an improved version of a recent theorem of Helfgott [8] and Shalev [20] 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.
doi:10.1017/s0017089514000251 fatcat:powajrfr45aobkwcr5oecaylaq