## 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:

 300000 Other combinatorial set theory

### Keywords:

rigid Souslin trees; diamond; automorphism tower
Full Text:

### References:

 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.