[bibtex key=HamkinsThomas2000:ChangingHeights]

If 𝐺 is a centreless group, then 𝜏⁡(𝐺) denotes the height of the automorphism tower of 𝐺. We prove that it is consistent that for every cardinal 𝜆 and every ordinal 𝛼 <𝜆, there exists a centreless group 𝐺 such that (a) 𝜏⁡(𝐺) =𝛼; and (b) if 𝛽 is any ordinal such that 1 ≤𝛽 <𝜆, then there exists a notion of forcing 𝑃, which preserves cofinalities and cardinalities, such that 𝜏⁡(𝐺) =𝛽 in the corresponding generic extension 𝑉𝑃.