The Ultraproducts of Quasirandom Groups

TL;DR

This paper investigates the properties of ultraproducts of quasirandom groups, showing conditions under which they remain quasirandom or become minimally almost periodic, with applications in group theory and combinatorics.

Contribution

It generalizes the behavior of ultraproducts of quasirandom groups, including simple and semisimple groups, and explores their representations and applications.

Findings

Ultraproducts of simple groups are either finite simple or have trivial representations.

Ultraproducts of increasingly quasirandom groups can become minimally almost periodic.

Applications include results in triangle patterns and self-Bohrifying groups.

Abstract

In this paper, we shall prove that an ultraproduct of non-abelian finite simple groups is either finite simple, or has no finite dimensional unitary representation other than the trivial one. Then we shall generalize this result for other kinds of quasirandom groups. A group is called D- quasirandom if all of its nontrivial representations over the complex numbers have dimensions at least D. We shall study the question of whether a non-principal ultraproduct of a given sequence of quasirandom groups remains quasirandom, and whether an ultraproduct of increasingly quasirandom groups becomes minimally almost periodic (i.e. no non-trivial finite-dimensional unitary representation at all). We answer this question in the affirmative when the groups in question are simple, quasisimple, semisimple, or when the groups in question have bounded number of conjugacy classes in their cosocles (the intersection of all maximal normal subgroups), or when the groups are arbitrary products (not necessarily finite) of the groups just listed. We shall also present with an ultraproduct of increasingly quasirandom groups with a non-trivial one-dimensional representation. We also obtain some results in the case of semi-direct products and short exact sequences of quasirandom groups. Finally, two applications of our results are given, one in triangle patterns of quasirandom groups and one in self-Bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.