Quantitative Preservations of Ulam Stability-type Estimates

TL;DR

This paper investigates how certain stability properties of groups are preserved under extensions and embeddings, providing quantitative bounds and conditions for stability in both finite and infinite dimensions.

Contribution

It introduces new quantitative preservation results for Ulam stability under group extensions, especially with H"older continuous moduli, and explores stability in high-dimensional embeddings.

Findings

Quantitative preservation of asymptotic bounds under group extensions.

Partial results on stability preservation in infinite-dimensional groups.

Strong quantitative stability results in finite-dimensional cases.

Abstract

We show some preservation results of amenably extending strongly Ulam stable groups under mild decay assumptions, including quantitative preservation of asymptotic bounds under the assumption that the modulus of stability is H\"older continuous of exponent $s>\frac 1 2$ at 0, utilizing some simplistic integral estimates. Additionally, we show some partial results around inductive preservation of modulus bounds in infinite dimensions using these integral estimates, as well as strong quantitative preservation in the finite dimensional case. This implies the existence of $\mathfrak{U}$ uniformly stable existential closures among groups with sufficiently large Lipschitz estimates of any countable group. Finally, we show quantitative control preserving difficulty of approximation of maps over stable groups on diagonally embedding into higher dimensions.