A hyperplane of a cube complex $X$ is a connected component of taking an $(n-1)$-cube for each midcube of $X$ and identifiying midcubes along faces of adjacent $n$-cubes of $X$. 


If a group $G$ acts on a finite dimensional CAT(0) cube complex not cocompactly (with perhaps extra conditions), must there be an infinite number of orbits of hyperplanes $G \hat{\mathfrak{h}}\_{1}, G \hat{\mathfrak{h}}\_{2}, \ldots $ such that the stabilizers of these hyperplanes $ Stab(\hat{\mathfrak{h}\_{i}}) \neq \{ 1 \}$ are non-trivial?



Non-cocompact action on CAT(0) cube complex and hyperplane stabilizers
Actually, having finitely many orbits of hyperplanes is quite common. For instance:


**Proposition:** (Sageev) Let $G$ be a finitely generated group acting on a CAT(0) cube complex $X$. Then there exists a $G$-invariant convex subcomplex $Y \subset X$ containing only finitely many $G$-orbits of hyperplanes. 


*Sketch of proof.* Let $s\_1, \ldots, s\_n$ be generators of $G$ and $x\_0 \in X$ a vertex. Set $Y$ as the convex hull of the orbit $G \cdot x\_0$. The hyperplanes of $Y$ are exactly the hyperplanes of $X$ separating two vertices of $G \cdot x\_0$. Let $J$ be such a hyperplane. So there exist $g,h \in G$ such that $J$ separates $gx\_0$ and $hx\_0$. By translating, we may suppose that $g=1$. Write $h$ as a word of generators $r\_1 \cdots r\_k$. By looking at a path
$$[x\_0,r\_1x\_0] \cup [r\_1x\_0,r\_1r\_2 x\_0] \cup \cdots \cup [r\_1 \cdots r\_{k-1}x\_0, r\_1 \cdots r\_{k-1}r\_k x\_0]$$
from $x\_0$ and $hx\_0$, we find that there exists some $i$ such that $J$ separates $r\_1 \cdots r\_ix\_0$ and $r\_1 \cdots r\_ir\_{i+1} x\_0$. Up to translating $J$, we may suppose that $J$ separates $x\_0$ and $r\_{i+1}x\_0$. 


In other words, any hyperplane of $Y$ has a translate which separates $x\_0$ from $s x\_0$ for some generator $s$. It follows that there exist only finitely many orbits of hyperplanes in $Y$. $\square$


In other words, if we are looking at finitely generated groups only, we can always suppose that the cube complex contains only finitely many orbits of hyperplanes, without any assumption on the action.


