Quasi-Random profinite groups

Abstract

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of ${\mathrm{SL}}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and ${\mathrm{Sp}}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. 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 ${\mathrm{SL}}_k({\mathbb{Z}}_p)$ and ${\mathrm{Sp}}_{2k}({\mathbb{Z}}_p)$. We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

arXiv commentThis is the final version. To appear in Glasgow Mathematical Journal