Coherence for elementary amenable groups
Sam Hughes, Dawid Kielak, Peter H. Kropholler, Ian J. Leary
TL;DR
The paper investigates coherence, homological coherence, and coherence of the group ring for elementary amenable groups, extending Bieri–Strebel's soluble-case equivalence to a broader class. It introduces a radical framework ${\mathbf L}\mathcal{P}$ and cascading groups, and develops indicable coherence alongside a transfinite elementary amenable hierarchy to control ascending HNN extensions. The main result proves that for finitely generated elementary amenable $G$, coherence, homological coherence, and coherence of $\mathbb{Z}G$ are equivalent to $G$ being either virtually polycyclic or a properly ascending HNN extension with a virtually polycyclic vertex group. It also establishes that Noetherian elementary amenable groups are virtually polycyclic and discusses Baer-type conjectures about Noetherianity and coherence of group rings in amenable contexts. Overall, the work generalizes core coherence phenomena from soluble groups to the elementary amenable setting and clarifies the link between ring-theoretic and group-theoretic coherence via hierarchical and HNN-extension techniques.
Abstract
We prove that for an elementary amenable group, coherence of the group, homological coherence of the group, and coherence of the integral group ring are all equivalent. This generalises a result of Bieri and Strebel for finitely generated soluble groups.