the Švarc-Milnor lemma
/ SHVARTS MILL-nor /
Here is the foundational miracle of geometric group theory: if a group acts on a space the way the symmetries of a tiling act on the plane — moving things around freely enough and tightly enough — then the group, as an abstract object, has the same large-scale shape as the space it acts on. The Švarc-Milnor lemma makes this precise. It says a group acting 'geometrically' on a nice space is quasi-isometric to that space. It is what lets you read a group's coarse geometry off any space it lives on, with no generating set in sight.
The setup: let G be a group acting by isometries on a length space X that is proper (closed balls compact) and geodesic. Call the action geometric — equivalently 'properly discontinuously and cocompactly' — if two conditions hold: (1) cocompact, meaning the quotient X/G is compact, so the orbit of any point is coarsely the whole space (there is a compact set whose G-translates cover X), and (2) properly discontinuous, meaning for any compact set K only finitely many group elements g have gK meeting K, so the action does not pile up. The conclusion: G is finitely generated, and for any basepoint x in X the orbit map g -> g.x, from G with a word metric to X, is a quasi-isometry. So G and X are the same at large scale.
This lemma is the reason geometry says anything about algebra at all. The fundamental group of a compact Riemannian manifold acts geometrically on the universal cover, so pi_1 is quasi-isometric to the universal cover — for example pi_1 of a compact hyperbolic surface is quasi-isometric to the hyperbolic plane, which is how you know surface groups are hyperbolic. It also proves the word metric is well-defined up to quasi-isometry (apply it to a group acting on its own Cayley graph). The hypotheses are not decorative: properness and cocompactness together are essential. Drop cocompactness and the orbit may be a thin sliver of X, giving no quasi-isometry; drop proper discontinuity and the group need not even be discrete. The action must be both spread out everywhere (cocompact) and discrete (properly discontinuous).
The group Z^2 acts on the plane R^2 by integer translations. This action is by isometries, it is cocompact (the quotient is a torus, which is compact, and the unit square's translates cover the plane), and it is properly discontinuous (only finitely many integer vectors move a bounded set to overlap itself). By Švarc-Milnor, Z^2 is quasi-isometric to R^2 — and indeed the word metric on Z^2 (the taxicab metric) is quasi-isometric to the Euclidean plane. The discrete group and the continuous plane share one large-scale geometry.
Z^2 acting on R^2 by translations is geometric, so Z^2 and R^2 are quasi-isometric.
Both hypotheses are load-bearing. Without cocompactness the orbit can miss most of the space; without proper discontinuity the group need not be discrete or finitely generated. A merely isometric action is far from enough — it must be geometric.