Quasi-Random profinite groups

TL;DR

This paper explores the quasi-randomness properties of profinite groups, providing bounds on representation degrees and measure bounds for product-free subsets, using techniques from algebra and functional analysis.

Contribution

It introduces new bounds for representation degrees and measure estimates for profinite groups, advancing understanding of their quasi-random properties.

Findings

Bounds for minimal degrees of non-trivial representations.

Exponential bounds for measure of product-free subsets.

Lower bounds for faithful representation degrees.

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.