On the rigidity of Souslin trees and their generic branches

Abstract

We show it is consistent that there is a Souslin tree $S$ such that after forcing with $S$, $S$ is Kurepa and for all clubs $C\subset {\omega }_1$, $S\upharpoonright C$ is rigid. This answers Fuchs's questions in Club degrees of rigidity and almost Kurepa trees. Moreover, we show it is consistent with $♢$ that for every Souslin tree there is a dense $X\subset S$ which does not have a copy of $S$. This is related to a question due to Baumgartner.