Characters of diagonal products and Hilbert-Schmidt stability

TL;DR

This paper introduces a new way to measure Hilbert-Schmidt stability in infinitely presented groups, demonstrating the existence of many stable amenable groups with large stability growth, and classifying characters of certain group families.

Contribution

It develops the concept of stability radius growth for Hilbert-Schmidt stability and classifies characters of specific group constructions, answering an open question.

Findings

Existence of uncountably many Hilbert-Schmidt stable amenable groups with arbitrarily large stability growth

Classification of characters for alternating, elementary enrichments, and diagonal products

Resolution of a question posed by Lubotzky

Abstract

We initiate a quantitative study of Hilbert-Schmidt stability for infinitely presented groups through the novel notion of stability radius growth. We exhibit an uncountable family of Hilbert-Schmidt stable amenable groups with arbitrarily large such growth. In particular, this answers a question of Lubotzky. Our approach is based on the character-theoretic stability criterion of Hadwin and Shulman. We classify the characters of alternating and elementary enrichments as well as diagonal products, including the classical family of B.H. Neumann groups.