Quasi-Random Profinite Groups

Abstract

Abstract. Inspired by Gowers' seminal paper [6], we will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of SL k (Z/(p n Z)) and Sp 2k (Z/(p n Z)). 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 SL k (Zp) and Sp 2k (Zp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

Effective and Infinite-Rank Superrigidity in the Context of Representation Stability

“…• There are similar dimension bounds for representations of Sp 2n (Z/p k Z) (again found in [21] and [2]).…”

Some remarks on the asymmetric sum-product phenomenon

“…In the proof he used the fact that equation (3) can be interpreted as a question about points/lines incidences. Clearly, the result above has the sum-product flavour and indeed one can use (3) to derive some lower bounds for the maximum from (2) (in the case of large subsets of Z q , of course).…”

“…It turns out that representation theory makes it possible to obtain (almost automatically) asymptotic formulae for the number of solutions to systems of equations that are preserved by the actions of certain groups. For example, equation (3) ). The advantage of our approach is its generality and (relative) simplicity.…”

Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields

“…Motivated by the works of Bourgain and Gamburd mentioned above, the first and third authors of this paper studied m f (SL k (Z/p n Z)) and m f (Sp 2k (Z/p n Z)) [1]. The same problem for m f (SL k (Z/p n Z)) has been considered by de Saxcé [8].…”

“…1. From now on, we fix a total ordering ≺ of Φ which is compatible with the height function ht, i.e.…”