Characters of solvable groups, Hilbert-Schmidt stability and dense periodic measures

TL;DR

This paper explores the character theory of specific solvable groups to analyze their Hilbert-Schmidt stability, revealing connections with automorphism dynamics and establishing stability for various classes of groups.

Contribution

It introduces a character-theoretic criterion for Hilbert-Schmidt stability and applies it to a broad class of solvable groups, linking stability with automorphism dynamics.

Findings

Finitely generated virtually nilpotent groups are Hilbert-Schmidt stable.

Free metabelian groups are Hilbert-Schmidt stable.

Lamplighter groups and certain upper triangular groups are Hilbert-Schmidt stable.

Abstract

We study the character theory of metabelian and polycyclic groups. It is used to investigate Hilbert-Schmidt stability via the character-theoretic criterion of Hadwin and Shulman. There is a close connection between stability and dynamics of automorphisms of compact abelian groups. Relying on this, we deduce that finitely generated virtually nilpotent groups, free metabelian groups, lamplighter groups as well as upper triangular groups over certain rings of algebraic integers are Hilbert-Schmidt stable.