Nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ in Cohen models

Abstract

We show that if $\kappa < \aleph_\omega$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ in the extension. Under some further hypotheses on the ground model, namely the existence of long enough sage Davies trees (which follows from $\mathsf{SCH}$ plus $\square_\lambda$ for every $\lambda$ with $\mathrm{cf}(\lambda) = \omega$), we prove the same result for cardinals $\kappa \geq \aleph_\omega$ as well. This extends a result a Shelah and Stepr\={a}ns, who proved the result for $\kappa = \aleph_2$.