Skip to main content

Degrees of rigidity for Souslin trees

  • Gunter Fuchs (a1) (a2) and Joel David Hamkins (a2) (a3)

We investigate various strong notions of rigidity for Souslin trees, separating them under ⟡ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ⟡ that there is a group whose automorphism tower is highly malleable by forcing.

Hide All
[1]Abraham, Uri, Construction of a rigid Aronszajn tree, Proceedings of the American Mathematical Society, vol. 77 (1979), no. 1, pp. 136137.
[2]Abraham, Uri and Shelah, Saharon, Isomorphism types of Aronszajn trees, Israel Journal of Mathematics, vol. 50 (1985), pp. 75113.
[3]Devlin, Keith J. and Johnsbråten, Havard, The Souslin problem, Lecture Notes in Mathematics, no. 405, Springer, Berlin, 1974.
[4]Fuchs, Gunter and Hamkins, Joel David, Changing the heights of automorphism towers by forcing with Souslin trees over L, this Journal, vol. 73 (2008), no. 2, pp. 614633.
[5]Gaifman, Haim and Specker, E. P., Isomorphism types of trees, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 16.
[6]Hamkins, Joel David, Every group has a terminating transfinite automorphism tower, Proceedings of the American Mathematical Society, vol. 126 (1998), no. 11, pp. 32233226.
[7]Hamkins, Joel David, How tall is the automorphism tower of a group?, Logic and algebra, AMS Contemporary Mathematics Series, vol. 302, 2001, pp. 4957.
[8]Hamkins, Joel David and Thomas, Simon, Changing the heights of automorphism towers, Annals of Pure and Applied Logic, vol. 102 (2000), no. 1-2, pp. 139157.
[9]Hulse, J. A., Automorphism towers of polycyclic groups, Journal of Algebra, vol. 16 (1970), pp. 347398.
[10]Jech, Thomas, Automorphisms of ω1-trees, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 5770.
[11]Jech, Thomas, Forcing with trees and ordinal definability, Annals of Mathematical Logic, vol. 7 (1974), pp. 387409.
[12]Jech, Thomas, Set theory, 3rd ed., Springer Monographs in Mathematics, 2003.
[13]Jech, Tomáš, Non-provability of Souslin's hypothesis, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 291305.
[14]Kurepa, G., Ensembles ordonnées et ramifiés, Publications de l'Institut Mathématique, Univ. Belgrade, vol. 4 (1935), pp. 1138.
[15]Rae, Andrew and Roseblade, James E., Automorphism towers of extremal groups, Mathematische Zeitschrift, vol. 117 (1970), pp. 7075.
[16]Scharfenberger-Fabian, Gido, Subalgebras of small Souslin algebras and maximal chains in Souslin algebras, Dissertation, Freie Universität Berlin, 03 2008.
[17]Tennenbaum, S., Souslin's problem. Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 6063.
[18]Thomas, Simon, The automorphism tower problem, Proceedings of the American Mathematical Society, vol. 95 (1985), pp. 166168.
[19]Thomas, Simon, The automorphism tower problem II, Israel Journal of Mathematics, vol. 103 (1998), pp. 93109.
[20]Thomas, Simon, The automorphism tower problem, to appear.
[21]Todorčević, Stevo, Rigid Aronszajn trees, Publications de l'Institut Mathematique (Beograd), Nouvelle Série, vol. 41 (1980), no. 27, pp. 259265.
[22]Wielandt, H., Eine Verallgemeinerung der invarianten Untergruppen, Mathematische Zeitschrift, vol. 45 (1939), pp. 209244.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed