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

Abstract

We show that if $\hat{I}º<{\aleph }_{\ddot{I}}‰$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal{P}(\ddot{I}‰)/\mathrm{Fin}$ 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 ${\square }_{\hat{I}}»$ for every $\hat{I}»$ with $\mathrm{cf}(\hat{I}»)=\ddot{I}‰$), we prove the same result for cardinals $\hat{I}º\ge {\aleph }_{\ddot{I}}‰$ as well. This extends a result a Shelah and Steprāns, who proved the result for $\hat{I}º={\aleph }_2$.