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Degrees of rigidity for Souslin trees. (English) Zbl 1179.03043

A Souslin tree \(T\) is (a) rigid if there is no nontrivial automorphism of \(T\), (b) totally rigid if whenever \(p\) and \(q\) are distinct nodes in \(T\), the subtrees \(\{ r\in T : p\leq_T r\}\) and \(\{ s\in T : q\leq_T s\}\) are not isomorphic, (c) UBP if \(1\Vdash_T T\) has exactly one new cofinal branch, (d) absolutely rigid if \(1\Vdash_T T\) is rigid, (e) absolutely totally rigid if \(1\Vdash_T T\) is totally rigid, and (f) absolutely UBP if \(\Vdash_T T\) is UBP. The authors observe the following implications between these properties : (f) \(\Rightarrow\) (c) \(\Rightarrow\) (b) \(\Rightarrow\) (a), (f) \(\Rightarrow\) (e) \(\Rightarrow\) (d) \(\Rightarrow\) (a) and (b)\(\Rightarrow\) (e). Moreover, they prove that the resulting diagram is complete. Indeed if \(\lozenge\) holds, then there are Souslin trees exhibiting each of the non-implications implicit in the diagram. Finally, as an application of their methods, the authors show that, assuming \(\lozenge\), there is, for each \(n<\omega\), a group \(G\) such that (i) the automorphism tower of \(G\) has height \(n\), and (ii) for \(0 < m < \omega\), there is a ccc forcing extension in which the automorphism tower of \(G\) has height \(m\).

MathOverflow Questions:

\(\omega_2\)-sequence of Suslin trees

MSC:

03E05 Other combinatorial set theory
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References:

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