Let $X$ be a CAT(0) cube complex.

(From Sageev and Wise's *Cores for Quasiconvex actions*)
A group $G$ acts *cosparsely* on a CAT(0)-cube complex $X$ if there exists a compact space $K$ and finitely many *quasiflats* $F_{1} , \ldots , F_{r}$ each quasi-isometric to $\mathbb{E}^{m}$ for some $m$ such that

- $X = GK \cup_{i} GF_{i}$
- Each hyperplane in $X$ crosses $GK$.
- $hF_{i} \cap kF_{j} \subset GK$ unless $i=j$ and $k^{-1}h \in $ Stabiliser$(F_{i})$
- Quasiflats are $D$-isolated in the sense that $hF_{i} \cap kF_{j}$ has diameter $< D$ unless $hF_{i} = kF_{j}$.

(From Caprace and Sageev's *Rank Rigidity of CAT(0)-cube complexes*)

Let $\Gamma $ be a group acting on $X$.
A hyperplane $\hat{\mathfrak{h}}$ of $X$ is called $\Gamma$-essential if both halfspaces contain $\Gamma$-orbit points of any vertex $v$ arbitrarily far away from $\hat{\mathfrak{h}}$.
An action on a CAT(0)-cube complex is called *essential* if every hyperplane is $\Gamma$-essential.

Are cosparse actions essential actions?
Sageev and Wise's *Cores for quasiconvex actions* (Prop 7.4) states that when a group $G$ is hyperbolic relative to virtually-free abelian groups, a proper and cosparse action on a CAT(0)-cube complex $X$ can be reduced to $G$ acting properly and cocompactly on a convex subspace of $X$ (which may not be a subcomplex and with convexity with respect to the CAT(0)-metric).

Lemma 3.1 of Caprace and Sageev's *Rank Rigidity for CAT(0)-cube complexes* implies that if $X$ strictly contains a $\Gamma$-invariant convex subcomplex, then the action of $\Gamma$ is not essential.