- 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://jdh.hamkins.org/changingheightsoverl/}, 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 $V^{P}$.

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Can we know anything more about those Groups, or about Heights of Automorphism Groups of known Groups?