  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.
