I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their stabilizers under the action of a hyperbolic group on a CAT(0) cube complex:

**Claim:** Suppose $G$ is a (word-)hyperbolic group acting properly discontinously cocompactly and faithfully on a $CAT(0)$ cube complex $X$. Then there exists an $R>0$ so that if two walls(hyperplanes) $W,W' \subset X$ are such that $d(W,W')>R$ then $G_W \cap G_{W'}$ is finite ($G_W \subset G$ denotes the stabilizer of $W$ i.e the elements that send $W$ to itself)

**Note:** Certain easy examples show that the condition that $G$ be hyperbolic is necessary, e.g $\mathbb{Z}^2$ acts on the cube complex $\mathbb{R}^2$ and the stabilizer of any two horizontal hyperplanes is $\mathbb{Z}$ which is infinite. A similair example shows that cocompactness is also a necessary assumption.