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

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.

• I hope Michah Sageev is well and you can ask him directly.
– user6976
Apr 18 '20 at 16:51
• @Carol: Any geometric action on a CAT(0) cube complex is cosparse, but it may not be essential. For instance, if $G$ acts on $X$ then the action $G \curvearrowright X \times [0,1]$ is not essential. But a reasonnable statement is that a cosparse action can always be made essentialy by extracting an invariant convex subcomplex. Apr 19 '20 at 6:11
• The case I'm more interested in is when the action is non-cocompact. Apr 19 '20 at 9:23

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.