Changing the heights of automorphism towers

Abstract

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 $1 \leq β< λ$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $τ(G) = β$ in the corresponding generic extension $V^{P}$.