Dimensions of finitely generated simple groups and their subgroups

Abstract

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of subgroups of a finitely generated simple torsion-free group. This is new even for basic cases such as $\{ 0, 1, 3 \}$ and $\{ 0, 1, \infty \}$, even without simplicity or finite generation, and answers a question of Talelli and disproves a conjecture of Petrosyan. Moreover, we prove that every countable group of dimension at least $2$ embeds into a finitely generated simple group of the same dimension. These are the first examples of finitely generated simple groups with dimension other than $2$ or $\infty$. As another application, we exhibit the first examples of torsion-free groups with the fixed point property for actions on finite-dimensional contractible CW-complexes, and construct torsion-free groups in all countable levels of Kropholler's hierarchy, answering a question of Januszkiewicz, Kropholler and Leary. Our method combines small cancellation theory with group theoretic Dehn filling, and allows to do several other exotic constructions with control on the dimension. Along the way we construct the first uncountable family of pairwise non-measure equivalent finitely generated torsion-free groups.