An approach to showing hyperbolic groups are CAT(0)

Question

I've been sitting on this idea for quite a while but I'm not in academia any longer so not likely to ever tackle it on my own. The approach is as follows:

1. $G$ acts on its boundary $\partial G$
2. ergo, $G$ also acts on the set of distinct, ordered triples of $\partial G$, call this $\Delta G$
3. $\Delta G$ factors as a hyperbolic space $\mathcal{H}G$ which is quasi-isometric to $G$, and some other (possibly compact) space, $\mathcal{C}G$
4. The $G$ action on $\Delta G$ "plays nice" so that there's a geometric $G$-action on $\mathcal{H}G$. Tada!

The motivation behind the approach is pretty simple: $\Delta G$ is the set of triangles with points on $\partial G$. Because $G$ is hyperbolic, those triangles can be readily identified with their centers which are coarsely elements of $G$ (this is why $\mathcal{H}G \sim G$) together with some other component that can be thought of as rotations of those triangles ($\mathcal{C}G$). $\Delta G$ is like the set of 2-frames on $G$, or something similar.

The distinction between $\mathcal{H}G$ and $G$ is that, by reconstructing everything from the boundary, $\mathcal{H}G$ has completely lost all local structure of $G$ that might be an impediment to CAT(0) (or CAT(-k), in fact). The details of how to factor $\Delta G$ are the biggest stumbling blocks I think.

In any case, I wanted to share in case this idea actually has promise to advance this question, or at least provide an avenue to identify more specific obstructions to the $CAT(0)$ condition.

To get started, are there any known techniques for recognizing factors in spaces like these, perhaps in the context of tangent bundles etc?

Take care,

Brad

Answer 1

This approach is quite hopeless for several reasons. First of all, let me try to make sense of what you wrote.

You write:

$\Delta 𝐺$ factors as a hyperbolic space ${\mathcal H}𝐺$ which is quasi-isometric to $𝐺$, and some other (possibly compact) space, ${\mathcal C} G$

So far, $\Delta 𝐺$ is just a topological space (yes, it has some incomplete metrics coming from Gromov-type metrics on $\partial G$, but these are not even $G$-invariant), so I read "factors" as "splits as a topological direct product."

You are hoping that the splitting is such that the $G$-action projects to a proper action on ${\mathcal H}𝐺$ and that the latter can be given an invariant $CAT(0)$ metric. The latter would imply contractibility and local contractibility.

But in "most" cases $\Delta 𝐺$ has rich topology. For instance, for generic $G$, $\partial G$ is homeomorphic to the Menger curve which has nontrivial $H^1$ even locally. Hence, by Kunneth, $\Delta 𝐺$ locally has nontrivial $H^3$. Hence, again by Kunneth, it cannot even locally split as a product of a contractible space (of positive dimension) with some other factor! Thus, there is no hope even for a local splitting (in general).

Edit. [Per Henry's request.] Below, all cohomology is Chech with rational coefficients (any field will work) and dimension means rational cohomological dimension:

Suppose that $A$ is a compact (and Hausdorff) 1-dimensional topological space (think of the Menger curve), $U_i\subset A, i=1, 2, 3$ is are open subsets with $H^1(U_i)\ne 0$. Then by the Kunneth formula, $H^3(W)\ne 0$, where $W=U_1\times U_2\times U_3$. Furthermore, $A^3$ has dimension 3.

Now, assume that all three subsets $U_i$ are such that $W$ is disjoint from the big diagonal in $A^3$, i.e. $W\subset Z:=A^{(3)}$, where the letter is the complement to the big diagonal in $A^3$.

From this, using the LES of the pair $(Z,W)$ and the fact that $H^4(Z)=0$, you see that $H^3(Z)\ne 0$. I will use it to prove that $Z=A^{(3)}$ does not split off a nontrivial contractible factor. (A similar argument works also locally.) Suppose to the contrary that $Z=B\times C$, where $C$ is contractible and $dim(C)>0$. Thus, $dim(B)+dim(C)=3$. In particular, $dim(B)\le 2$ and, hence, $H^3(B)=0$. By applying the Kunneth formula again, we get $$ H^3(Z)= H^3(B)\otimes H^0(C)= H^3(B)=0. $$ A contradiction.

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Even if you assume that $\partial G$ is a topological $n$-sphere, your approach is just a "naive approach to Cannon's conjecture." The problem is that you know that $\Delta 𝐺$ is homeomorphic to the product of a compact with $R^{n+1}$. However, this product decomposition cannot be $G$-invariant. You can only hope to get a $G$-bundle over $R^{n+1}$. Nobody succeeded (so far) in finding such bundles directly for any $n>0$. The known cases are $n=1$ and $n\ge 5$. All require very hard and nontrivial work by first-rate topologists (Gabai, Casson, Weinberger...). The proof by Casson and Jungreis is closest to the idea of finiding an equivariant fibration, but it is based on some very special 3-dimensional topology tools.

Lastly, even if you are given a $G$-bundle over a contractible space $X$, how do you propose to find a $G$-invariant CAT(0)-metric on $X$? As an exercise, consider the case of 3-manifold groups. So, you got a proper and cocompact action of your hyperbolic group on a contractible 3-dimensional manifold $X$ (the base of a fibration of $\Delta G$). How do you know that there is a $G$-invariant CAT(0)-metric on $X$? Right, you will need to quote Perelman's theorem.

Edit. Few more things in this direction: What are the known tools for finding a locally CAT(0)-metric on the given topological space $Y$? There are only few:

a. Combinatorial, in case when $Y$ is a cell complex with a particularly nice local properties. Once $Y$ has sufficiently large dimension, all you have along these lines is the construction of (locally) CAT(0) cube complexes. This excludes hyperbolic groups with Property (T).

b. Some special differential-geometric constructions for manifolds of dimensions $\le 3$ and in the case of locally symmetric spaces.

c. Some variations on cut-and-paste or branched covering constructions using the pre-existing locally CAT(0) metrics.

My list is missing few more sporadic constructions but, I think, the situation is quite clear: Given a general locally contractible and aspherical compact topological space $Y$ (of sufficiently high dimension) with nontrivial hyperbolic fundamental group we simply lack any tools for constructing a locally CAT(0) metric on $Y$. Cf. this question regarding CAT(1) metrics.