Changing the heights of automorphism towers

  • J. D. Hamkins and S. Thomas, “Changing the heights of automorphism towers,” Ann.~Pure Appl.~Logic, vol. 102, iss. 1-2, pp. 139-157, 2000.  
    @article {HamkinsThomas2000:ChangingHeights,
    AUTHOR = {Hamkins, Joel David and Thomas, Simon},
    TITLE = {Changing the heights of automorphism towers},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {102},
    YEAR = {2000},
    NUMBER = {1-2},
    PAGES = {139--157},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {20F28 (03E35 20A15)},
    MRNUMBER = {1732058 (2000m:20057)},
    MRREVIEWER = {Markus Junker},
    DOI = {10.1016/S0168-0072(99)00039-1},
    URL = {http://dx.doi.org/10.1016/S0168-0072(99)00039-1},
    eprint = {math/9703204},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

If $G$ is a centreless group, then $\tau(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $\lambda$ and every ordinal $\alpha < \lambda$, there exists a centreless group $G$ such that (a) $\tau(G) = \alpha$; and (b) if $\beta$ is any ordinal such that $1 \leq \beta < \lambda$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $\tau(G) = \beta$ in the corresponding generic extension $V^{P}$.