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.