Automorphisms of $\omega _{1}$-trees

Abstract:

The number of automorphisms of a normal ${\omega _1}$-tree $T$, denoted by $\sigma (T)$, is either finite or ${2^{{\aleph _0}}} \leqslant \sigma (T) \leqslant {2^{{\aleph _1}}}$. Moreover, if $\sigma (T)$ is infinite then $\sigma {(T)^{{\aleph _0}}} = \sigma (T)$. Moreover, if $T$ has no Suslin subtree then $\sigma (T)$ is finite or $\sigma (T) = {2^{{\aleph _0}}}$ or $\sigma (T) = {2^{{\aleph _1}}}$. It is consistent that there is a Suslin tree with arbitrary precribed $\sigma (T)$ between ${2^{{\aleph _0}}}$ and ${2^{{\aleph _1}}}$, subject to the restriction above; e.g. ${2^{{\aleph _0}}} = {\aleph _1},{2^{{\aleph _1}}} = {\aleph _{324}}$ and $\sigma (T) = {\aleph _{17}}$. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen’s method of forcing and Jensen’s techniques in $L$.