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

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.