Conjugating trivial automorphisms of $\mathcal P(\mathbb N)/\mathrm{Fin}$

TL;DR

This paper investigates conditions under which trivial automorphisms of the Boolean algebra $ ext{P}( ext{N})/ ext{Fin}$ are conjugate, linking set-theoretic axioms, forcing extensions, and the structure of automorphisms.

Contribution

It establishes the equivalence between conjugacy of trivial automorphisms and the absence of obstructions under CH, and computes their associated structures' existential theories.

Findings

Under CH, conjugacy corresponds to absence of obstructions.

Automorphisms are conjugate in some forcing extension.

Existential theories of associated structures are computed.

Abstract

A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) cycle structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are neither first-order obstructions nor index obstructions for their conjugacy. This is equivalent to given trivial automorphisms being conugate in some forcing extension of the universe. To each automorphism $\alpha$ of $\mathcal P(\mathbb N) / \mathrm{Fin}$ we associate the first-order structure $\mathfrak{A}_\alpha=(\mathcal P(\mathbb N) / \mathrm{Fin},\alpha)$ and compute the existential theories of these structures. These results are applied to resolve a question of Braga, Farah, and Vignati and prove that there are coarse metric spaces $X$ and $Y$ such that the isomorphism between their uniform Roe coronas is independent from $\mathsf{ZFC}$.