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).