Forcing Over a Free Suslin Tree

Abstract

We introduce a forcing for adding almost disjoint automorphisms of a normal infinitely splitting $\ddot{I}{‰}_1$-tree $T$ with countable approximations. Assuming that $T$ is a free Suslin tree, this forcing is totally proper, preserves the Suslinness of $T$, and does not add new cofinal branches of $\ddot{I}{‰}_1$-trees existing in intermediate extensions. If $\hat{I}º$ is an inaccessible cardinal, then the product of the automorphism forcing of length $\hat{I}º$ with the Lévy collapse of $\hat{I}º$ to become $\ddot{I}{‰}_2$ forces that there exists an almost Kurepa Suslin tree and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Jin, Shelah, and Moore.

To appear in Advances in Mathematics