For instance, if an automorphism of a free group doesn’t fix any conjugacy classes then the free-by-cyclic group is hyperbolic, but in general I don’t know a reason why it should be CAT(-1).

Similarly, any group whose relations satisfy the small-cancellation condition is hyperbolic, but again I don’t think a reason is known for such a group to be CAT(-1). (See Lyndon & Schupp for the definition of .)

The last example can be generalised to Gromov’s random groups, which are (with overwhelming probability) hyperbolic whenever they are infinite. Yet again, I don’t think any reason is known for such a group to be CAT(-1). (Though in some small ranges they are known to be CAT(0), by work of Ollivier and Wise.)

]]>The approach you describe is the one taken by Hamish/Noel, except the business of local-geodesics being quasi-geodesics and quasi-geodesics being close to geodesics is bypassed… or perhaps, more accurately, is nicely distilled to the bare essence that is need to prove the result.

]]>I like this proof because it’s a nice example of the sort of local-to-global argument that shows up everywhere when you talk about hyperbolic groups. Basically, one version (I’m not sure if this is Noel’s version or not) goes like this:

Let $K$ be a sufficiently large number. Let $w:[0,l(w)]\to C_\Gamma(A)$ be a loop.

Say that there are $0\le x,y\le l(w)$ such that $|x-y|\le K/2$ and $d(w(x),w(y))< |x-y|$. Then we can shorten $w$ by replacing that segment of $w$ with a geodesic. If every sufficiently large loop can be shortened this way, that means that the group has a linear Dehn function — this reduction process corresponds to a decomposition into loops of length $K$ (this property that every loop can be shortened is what it means for the group to have a Dehn presentation).

So, say that $w$ is a loop which cannot be shortened like this. Then $w$ is a $K/2$-local geodesic (that is, a curve such that if $|x-y|\le K/2$, then $d(w(x),w(y)) = |x-y|$). It turns out that $K/2$-local geodesics are in fact quasi-geodesics when $K$ is sufficiently large (the local-to-global argument I mentioned). This takes some work, but the main tool is the fact that quasi-geodesics are close to geodesics. So now, $w$ is a curve which is within a neighborhood of the geodesic between its endpoints, but that geodesic is just the constant curve, so $w$ has bounded length and the group has a linear Dehn function.

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