Is a cosparse action on a CAT(0) cube complex an essential action?

Question

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

1. $X = GK \cup_{i} GF_{i}$
2. Each hyperplane in $X$ crosses $GK$.
3. $hF_{i} \cap kF_{j} \subset GK$ unless $i=j$ and $k^{-1}h \in $ Stabiliser$(F_{i})$
4. 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.

Answer 1

Being a cosparse action is not a rigid property. If $G \curvearrowright X$ is any cosparse action on a CAT(0) cube complex, then $G\curvearrowright X \times [0,1]$ is again cosparse but it is not essential. For instance, any action of a finite group on a CAT(0) cube complex which is not a single vertex is cosparse but not essential.

If you are interested in less trivial example, consider the action of $\mathbb{Z}$ on $\mathbb{R}^2 \times [0,1]$ through the translation by $(1,1,0)$. The action is already cosparse but not essential (nor cocompact). But you can make the example more complicated. The quotient $Q:=(\mathbb{R}^2 \times [0,1]) / \mathbb{Z}$ is a product of $[0,1]$ with a non-positively curved square complex which is topologically a cylinder and which has two hyperplanes. Consider the pointed sum $Q \vee \mathbb{S}^1$. Its fundamental group (a free group $\mathbb{F}_2$) acts on its universal cover (a CAT(0) cube complex which is a tree of quasiflats $\mathbb{R}^2 \times [0,1]$), and this action is cosparse but not essential.