Seeking an Application of Svarc-Milnor Lemma

Question

The Svarc-Milnor lemma is the following: Let $X$ be a length space*, and $\Gamma$ a group acting properly and cocompactly by isometries on $X$. Then $\Gamma$ is finitely generated.

This is an interesting and useful lemma in metric geometry. I would like to find a direct application of this lemma to group theory. That is to say, I would like to find an example of a space $X$ and a group $\Gamma$ which is not obviously finitely generated, where it is possible to show that $\Gamma$ has the necessary properties in order to actually prove that $\Gamma$ is finitely generated. If possible, I would like to work out a presentation for this group, but maybe this is too much to ask for.

After a few days of mulling it over, an example has eluded me. An example or reference would be wonderful!

*: A length space is defined as a space where the distance between every pair of points is equal to the infimum of the length of rectifiable curves joining the points. Any geodesic space is a length space. For an example of a metric space which is not a length space, consider $\mathbb Q \subset \mathbb R$, with the induced metric. There are no curves connecting points of $\mathbb Q$ while actually staying in $\mathbb Q$ - all curves are 'constant curves.' So there aren't even any curves joining any points for there to be any infimums to take.

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Answer 1

One example is in the paper "Morse theory and finiteness properties of groups" by Bestvina and Brady, Inventiones Mathematicae, 1997, Volume 129, Issue 3, pp 445–470.

In the paper they construct examples of finitely generated groups which are of type $FP_2$ but not finitely presented (which was an open problem for a long time). While finite generation of these groups is not the hardest part of the paper, it is not obvious. Finite generation follows from connectendness of certain complexes on which these groups are acting cocompactly (cocompactness is clear); connectedness, in turn, follows from some Morse-theoretic considerations. (The property $FP_2$ follows from acyclicity of these complexes.)

Answer 2

Let $q$ be the quadratic form $X^2+Y^2+(1+\sqrt{2})Z^2$. Define $\Gamma=SO(q)(\mathbf{Z}[\sqrt{2}]$. Then one shows that $\Gamma$ is finitely generated using its proper cocompact isometric action on the hyperbolic plane.

More generally, cocompact lattices in virtually connected Lie groups are finitely generated, by this argument.

Answer 3

You can use Svarc-Milnor Lemma to show that $Out(F_n)$ is f.g. using its action on the spine $\mathcal{K}_n \subset \mathcal{X}_n$.

See for example in Corollary 2.6.2 of "The topology, geometry, and dynamics of free groups" by Lee Mosher