Changing the heights of automorphism towers by forcing with Souslin trees over $L$

  • G. Fuchs and J. D. Hamkins, “Changing the heights of automorphism towers by forcing with Souslin trees over $L$,” J.~Symbolic Logic, vol. 73, iss. 2, pp. 614-633, 2008.  
    @ARTICLE{FuchsHamkins2008:ChangingHeightsOverL,
    AUTHOR = {Fuchs, Gunter and Hamkins, Joel David},
    TITLE = {Changing the heights of automorphism towers by forcing with {S}ouslin trees over {$L$}},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {73},
    YEAR = {2008},
    NUMBER = {2},
    PAGES = {614--633},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E35},
    MRNUMBER = {2414468 (2009e:03094)},
    MRREVIEWER = {Lutz Struengmann},
    URL = {http://dx.doi.org/10.2178/jsl/1208359063},
    doi = {10.2178/jsl/1208359063},
    eprint = {math/0702768},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F
    }

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

In an earlier paper with Simon Thomas, “Changing the heights of automorphism towers,”¬†we had added such malleable groups by forcing, and the current paper addresses the question as to whether there are such groups already in L.