Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite - MathOverflow

Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite

2012-12-08T08:47:38Z

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.

Answer by HW for Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite

2012-12-08T18:11:13Z

Here's a proof.

Lemma: Suppose $G$ is a (word-)hyperbolic group acting properly discontinously, cocompactly and faithfully on a C⁢A⁢T⁢(0) space $X$. Then there is a uniform bound $R_0$ on the width $R$ of isometrically embedded flat strips $\mathbb{R}\times [0,R]$ in $X$.

Proof: If not then, by cocompactness, there exist nested flat discs of diameter tending to infinity. Their union is an embedded copy of $\mathbb{R}^2$, which contradicts hyperbolicity. QED

Now suppose that walls $W,W'$ have stabilizers with infinite intersection. Then that intersection is an infinite word-hyperbolic group (since it is quasiconvex in $G$) and so contains an element $\gamma$ of infinite order. Because $W,W'$ are convex and so themselves CAT(0), each contains an axis $l_W,l_{W'}$ for $\gamma$. By standard facts about CAT(0) spaces (see Bridson--Haefliger), any two axes bound a flat strip. Therefore, by the lemma, $l_W$ and $l_{W'}$ are at distance at most $R_0$, as claimed.