Nontrivial automorphisms of P ( ω ) / Fin P(ω )/Fin script upper P left parenthesis omega right parenthesis divided by upper F i n in Cohen models

We show that if κ < ℵ ω κ < _ω kappa less than normal first transfinite cardinal Subscript omega Cohen reals are added to a model of CH CH sans serif CH , then there are nontrivial automorphisms of P ( ω ) / Fin P(ω )/Fin script upper P left parenthesis omega right parenthesis divided by upper F i n in the extension. Under some further hypotheses on the ground model, namely, the existence of long enough sage Davies trees (which follows from SCH SCH sans serif SCH plus ◻ λ _λ white medium square Subscript lamda Baseline for every λ λ lamda with cf ( λ ) = ω cf(λ ) = ω cf left parenthesis lamda right parenthesis equals omega ), we prove the same result for cardinals κ ≥ ℵ ω κ ≥ _ω