Universal flows and automorphisms of $\mathcal P(\omega)/\mathrm{fin}$

TL;DR

The paper constructs universal flows for any countable group on the Čech-Stone remainder of the natural numbers, and under CH, it shows the existence of a trivial automorphism embedding all others of \\mathcal P(\\omega)/fin.

Contribution

It introduces universal G-flows for countable groups on \\omega^* and demonstrates, under CH, a trivial automorphism of \\mathcal P(\\omega)/fin that embeds all automorphisms, advancing understanding of automorphism structures.

Findings

Existence of G-flows on \\omega^* with all smaller G-flows as quotients.

Under CH, existence of a universal G-flow of weight \\mathfrak{c}.

Under CH, a trivial automorphism of \\mathcal P(\\omega)/fin embedding all automorphisms.

Abstract

We prove that for every countable discrete group $G$, there is a $G$-flow on $\omega^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of weight $\leq\!\mathfrak{c}$. Applying Stone duality, we deduce that, under \mathsf{CH}, there is a trivial automorphism $\tau$ of $\mathcal P(\omega)/\mathrm{fin}$ with every other automorphism embedded in it, which means that every other automorphism of $\mathcal P(\omega)/\mathrm{fin}$ can be written as the restriction of $\tau$ to a suitably chosen subalgebra.