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Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:

There does not exist any Riemannian metric on the torus whose sectional curvature is $<0$.

This is a consequence of Milnor's article A note on curvature and fundamental group, where he proves that the fundamental group of a negatively-curved Riemannian manifold has exponential growth. On the other hand, the fundamental group of the torus, namely $\mathbb{Z}^2$, has quadratic growth.

Any compact Riemannian manifold whose sectional curvature is $\equiv 0$ has a torus as a finite cover.

This is a consequence of Bieberbach theorem.

More recently, showing that quasiconvex subgroups of hyperbolic cubulable groups are separable was the key point in the proof of the virtual Haken's conjecture. However, this is more technical.

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    $\begingroup$ I'm not sure whether I would call Milnor's theorem a "result proved by group theory". I think the proof is pure geometry. $\endgroup$
    – ThiKu
    Commented May 9, 2016 at 17:46
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    $\begingroup$ @ThiKu: The same is actually true for Bieberbach's theorem. It also doesn't really use group theory. $\endgroup$
    – eins6180
    Commented May 9, 2016 at 18:14
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    $\begingroup$ I think the virtual Haken and virtually Fibering conjectures are good examples where group theory plays a key role although there are other techniques as well. $\endgroup$
    – Benjamin Steinberg
    Commented May 9, 2016 at 19:57
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    $\begingroup$ I'm not convinced that any such results are proved purely using group theory: certainly none of the examples given so far qualify. The fact of the matter is that modern group theory is inextricably intertwined with geometry. If the question is just 'list theorems in geometric and topological group theory', then it's far too broad $\endgroup$
    – HJRW
    Commented May 9, 2016 at 20:32
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    $\begingroup$ For the proof of Milnor's theorem, I think one can phrase it without mentioning groups: any two metrics on a compact surface are quasi-isometric, so are their lifts to the universal covering space. So the volume growth of metric balls has to be of the same type (exponential or polynomial) in both cases. So one can not have both, a flat metric and a negatively curved one. The important concept here is that of quasi-isometry, while the fundamental group and the Milnor-Svarc lemma are actually not necessary. $\endgroup$
    – ThiKu
    Commented May 12, 2016 at 4:37

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Determining whether or not two simplicial complexes are homeomorphic is undecidable. Markov fils showed this in 1958 by reduction to the word problem for groups.

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    $\begingroup$ This is one of my favourite examples, but again it's not purely group-theoretic. Markov had to be careful to ensure that the simply-connected manifolds he constructed were homeomorphic. (These days it's easier using Freedman, but that's an even bigger topological hammer!) $\endgroup$
    – HJRW
    Commented May 10, 2016 at 5:49
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I'm still a little uncertain about this question, but I'll try to say something about the Virtual Haken conjecture (discussed above) and in the process explain why I think it's a good example.

The Virtual Haken conjecture (now Agol's theorem) can be stated as follows --

Every hyperbolic 3-manifold has a finite-sheeted cover that contains an embedded, incompressible surface.

-- a thoroughly topological statement (though more group-theoretic statements can be given).

For me, modern (geometric) group theory enters the picture in a truly astonishing way via the following result, Wise's Malnormal Special Quotient Theorem (MSQT).

Let $G$ be a hyperbolic, virtually special group, and $H$ a malnormal, quasiconvex subgroup. Then, for all sufficiently deep finite-index subgroups $K\lhd H$, the quotient $G/\langle\langle K\rangle\rangle$ is hyperbolic and virtually special.

I won't explain the definitions here, but rather point out that this tells us that we can kill large subgroups and stay in the (very well behaved) world of hyperbolic, virtually special groups. Note that this is not true of manifolds. You can't crush an immersed surface in a hyperbolic manifold and get a new manifold.

Although the proofs of the MSQT can (and usually are) phrased in an entirely topological/geometric manner, the key point here is that they concern geometric complexes (more precisely, CAT(0) cube complexes), not manifolds. The transition from manifolds to more general geometric complexes is surely the hallmark of modern infinite group theory.

Agol's proof, still the only proof we know, makes essential use of the MSQT. In this sense, the Virtual Haken conjecture is truly a theorem of geometric group theory rather than topology, in the sense that we don't know how to keep the proof purely in the world of topology (ie manifolds) -- you have to pass to the world of CAT(0) cube complexes (ie group theory).

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