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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.

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  • $\begingroup$ If the intersection is infinite it contains an infinite order element, whose axis lies in a neighbourhood of both walls. So the two walls bound a wide flat strip, which contradicts $\delta$-hyperbolicity. $\endgroup$
    – HJRW
    Dec 8, 2012 at 9:12
  • $\begingroup$ Pardon my ignorance, but I don't really follow. What sort of a neighbourhood of the walls are you considering? What is a "wide flat strip?". How do I find a global $R$ that works? Keep in mind that not all intersections are finite(only sufficiently far apart ones are) and this $R$ depends on $X$, for example consider $\mathbb{Z}$ acting by horizontal translations on the cube complex $\mathbb{R} \times [0,N]$, here you need $R \geq N-1$. $\endgroup$ Dec 8, 2012 at 10:01
  • $\begingroup$ I'll try to write down some details when I have time (Monday?), unless someone (eg Agol!) beats me to it. $\endgroup$
    – HJRW
    Dec 8, 2012 at 15:55
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    $\begingroup$ As HW says, this follows from basic properties of CAT(0) and hyperbolic spaces. The key property is: If $g_1, g_2$ are two infinite geodesics in $\delta$-hyperbolic space which are Hausdorff-close then there are points $x_1\in g_1, x_2\in g_2$ so that $d(x_1, x_2)\le 2\delta$. In your case, geodesics will be axes of hyperbolic isometries preserving the who walls and $g_1, g_2$ are their axes. A general suggestion is to read (at least a bit) of Bridson-Haefliger before reading Agol's paper. $\endgroup$
    – Misha
    Dec 8, 2012 at 16:42
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    $\begingroup$ I'm revising the paper now, so I'll try to add some exposition on this point. Either Henry or Misha's answers are what I had in mind. $\endgroup$
    – Ian Agol
    Dec 9, 2012 at 21:35

1 Answer 1

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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.

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