[bibtex key=HamkinsThomas2000:ChangingHeights]

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 $1\le \beta <\lambda$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $\tau (G)=\beta$ in the corresponding generic extension $V^P$.