21.36 (2026)

Kropholler’s hierarchy (see 15.45) is closed under finite extensions, that is, $(\mathbf{H}_\alpha \mathfrak{F})\mathfrak{F} \subseteq \mathbf{H}_\alpha \mathfrak{F}$ for every $\alpha$ (P. Kropholler, J. Pure Appl. Algebra, 90 (1993), 55–67). Let a hierarchy of tdlc groups $\mathbf{H} \mathfrak{K}$ be defined analogously to Kropholler’s hierarchy in 15.45, with $\mathfrak{K}$ being the class of profinite groups and with the cell stabilisers of the admissible action required to be open. Is it true that $\mathbf{H} \mathfrak{K}$ is closed under profinite extensions, that is, $(\mathbf{H}_\alpha \mathfrak{K})\mathfrak{K} \subseteq \mathbf{H}_\alpha \mathfrak{K}$ for every $\alpha$?