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


03E05 Other combinatorial set theory
Full Text: DOI arXiv


[1] DOI: 10.1007/BF01580283 · Zbl 0021.21003
[2] Transactions of the American Mathematical Society 173 pp 57– (1972) · Zbl 0247.00014
[3] DOI: 10.1016/0021-8693(70)90015-3 · Zbl 0207.03404
[4] DOI: 10.1016/S0168-0072(99)00039-1 · Zbl 0944.03048
[5] Logic and algebra 302 pp 49– (2001)
[6] DOI: 10.1090/S0002-9939-98-04797-2 · Zbl 0904.20027
[7] DOI: 10.1090/S0002-9939-1964-0168484-2
[8] Changing the heights of automorphism towers by forcing with Souslin trees over L 73 pp 614– (2008) · Zbl 1153.03026
[9] The Souslin problem (1974) · Zbl 0289.02043
[10] DOI: 10.1007/BF02761119 · Zbl 0566.03032
[11] DOI: 10.1090/S0002-9939-1979-0539646-8
[12] DOI: 10.1007/BF02762269 · Zbl 0919.20026
[13] DOI: 10.1090/S0002-9939-1985-0801316-9
[14] DOI: 10.1073/pnas.59.1.60 · Zbl 0172.29503
[15] DOI: 10.1007/BF01109829 · Zbl 0213.30101
[16] Publications de l’Institut Mathématique, Univ. Belgrade 4 pp 1– (1935)
[17] Commentationes Mathematicae Universitatis Carolinae 8 pp 291– (1967)
[18] Set theory (2003)
[19] Annals of Mathematical Logic 7 pp 387– (1974)
[20] Publications de l’Institut Mathematique (Beograd), Nouvelle Série 41 pp 259– (1980)
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