the Margulis lemma
/ mar-GOO-lis /
In a negatively curved space, if several symmetries each move a point only a tiny amount, you might expect them to interact tamely. The Margulis lemma makes this precise and surprisingly strong: there is a single universal distance, depending only on the dimension and the curvature bound, such that the symmetries which barely move any given point automatically generate an almost-abelian group. Below this threshold geometry is forced to be simple, and this is what cleaves a hyperbolic manifold into a tame 'thick' part and a tube-and-cusp 'thin' part.
Precisely, for each dimension n there is a Margulis constant eps(n) > 0 with the following property. Let M = H^n / Gamma be a complete hyperbolic n-manifold and pick any point x; consider the subgroup Gamma_eps(x) of Gamma generated by all elements g with d(x, g.x) < eps(n) (the isometries that move x less than eps). Then Gamma_eps(x) is virtually nilpotent — it contains a nilpotent subgroup of finite index. In dimension 3 such a group is either generated by a single hyperbolic element (giving a thin tube around a short closed geodesic) or is the parabolic stabilizer of a cusp (giving a cusp neighborhood, a quotient of a horoball).
The consequence is the thick-thin decomposition. Define the eps-thin part of M as the points x where some nontrivial loop through x has length < eps(n) (equivalently injectivity radius < eps/2); its complement is the thick part. Margulis guarantees every connected component of the thin part is a standard piece: a tube around a short geodesic or a cusp. This local-to-global control is the foundation of finiteness theorems (only finitely many hyperbolic 3-manifolds of bounded volume), of volume rigidity, and of much of the geometry behind geometrization.
On a closed hyperbolic 3-manifold, suppose a closed geodesic gamma has length 0.01, far below the Margulis constant. Then the thin part around gamma is an embedded solid-torus tube: every point inside it lies on a noncontractible loop shorter than eps, and the loops are all powers of gamma's core. As gamma's length tends to 0 the tube becomes infinitely long and thin and, in the limit, opens into a cusp. The thick part is what remains after removing all such tubes and cusps.
Thin part anatomy: a short geodesic sits in a tube; shrinking its length opens the tube into a cusp.
The Margulis constant is universal in the dimension but its exact value is not known in dimension 3 — explicit lower bounds exist, yet the optimal constant is open. Also, 'virtually nilpotent' is the right strength: the thin-part group is not literally abelian, only nilpotent up to finite index.