Cubulation of Hyperbolic Surface-by-Free Groups

Definitions and mathematical statement

A group is word-hyperbolic if it has a finite generating set whose Cayley graph has $\delta$-thin geodesic triangles for some $\delta\geq 0$. Write $F_r$ for the free group of finite rank $r\geq 0$. An exact sequence \[ 1\longrightarrow \pi_1(S)\xrightarrow{\iota}G\xrightarrow{p}Q\longrightarrow 1 \] means that $\iota$ is injective, $p$ is surjective, and $\ker p=\iota(\pi_1(S))$.

A $\operatorname{CAT}(0)$ cube complex is a complex obtained by gluing Euclidean cubes along faces whose intrinsic metric is geodesic and satisfies the $\operatorname{CAT}(0)$ comparison inequality. Here cubulable means admitting an action by cubical automorphisms on such a complex $X$ that is free, proper, and cocompact: no nonidentity element fixes a point; for every compact $K\subseteq X$, the set $\{g\in G:gK\cap K\neq\varnothing\}$ is finite; and $G\backslash X$ is compact.

Question. For every closed surface $S$ of genus greater than one and every exact sequence as above, if $G$ is word-hyperbolic and $Q\cong F_r$ for some finite $r\geq 0$, does there exist a $\operatorname{CAT}(0)$ cube complex $X$ on which $G$ acts freely, properly, and cocompactly?

The supplied statement does not explicitly specify whether surfaces must be connected and orientable. Its surface and genus terminology is retained here; these conventions require confirmation before the quantification over $S$ can be regarded as fully resolved. The fundamental group is understood with a chosen base point. No positive lower bound on the rank of $Q$ is supplied.

Short English statement

Must every word-hyperbolic group that is an extension of a closed surface group of genus greater than one by a finitely generated free group act freely, properly, and cocompactly on a $\operatorname{CAT}(0)$ cube complex? The source leaves the surface conventions implicit.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003945$, title Cubulation of Hyperbolic Surface-by-Free Groups, Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is identified. The supplied proposed year is 2018.
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, pp. 2475--2534, DOI 10.4171/owr/2018/40. https://doi.org/10.4171/owr/2018/40. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Mj--Manning--Sageev, Cubulating Surface-by-free Groups. https://arxiv.org/abs/1908.03545.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the cited work proves sufficient conditions and examples while explicitly leaving the general hyperbolic surface-by-free cubulability question open. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that mathematical content was preserved while notation, typography, and omitted definitions were repaired for self-contained publication. These are supplied editorial claims, not mathematical verification performed for this draft.