Margulis lemma

In differential geometry, a subfield of mathematics, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds (e.g. the hyperbolic n-space). Roughly, it states that within a fixed radius, usually called the Margulis constant, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup (in fact a bounded finite number of such).
The Margulis lemma for manifolds of non-positive curvature.
Formal statement.
The Margulis lemma can be formulated as follows.
Let formula_1 be a simply-connected manifold of non-positive bounded sectional curvature. There exist constants formula_2 with the following property. For any discrete subgroup formula_3 of the group of isometries of formula_1 and any formula_5, if formula_6 is the set:
then the subgroup generated by formula_6 contains a nilpotent subgroup of index less than formula_9. Here formula_10 is the distance induced by the Riemannian metric.
An immediately equivalent statement can be given as follows: for any subset formula_11 of the isometry group, if it satisfies that: 
then formula_14 contains a nilpotent subgroup of index formula_17.
Margulis constants.
The optimal constant formula_18 in the statement can be made to depend only on the dimension and the lower bound on the curvature; usually it is normalised so that the curvature is between -1 and 0. It is usually called the Margulis constant of the dimension.
One can also consider margulis constants for specific spaces. For example there has been an important effort to determine the Margulis constant of the hyperbolic spaces (of constant curvature -1). For example:
Zassenhaus neighbourhoods.
A particularly studied family of examples of negatively curved manifolds are given by the symmetric spaces associated to semisimple Lie groups. In this case the Margulis lemma can be given the following, more algebraic formulation which dates back to Hans Zassenhaus. 
Such a neighbourhood is called a Zassenhaus neighbourhood.
Thick-thin decomposition.
Let formula_31 be a Riemannian manifold and formula_32. The "thin part" of formula_31 is the subset of points formula_34 where the injectivity radius of formula_31 at formula_36 is less than formula_18, usually denoted formula_38, and the "thick part" its complement, usually denoted formula_39. There is a tautological decomposition into a disjoint union formula_40.
When formula_31 is of negative curvature and formula_18 is smaller than the Margulis constant for formula_43 the structure of the components of the thin part is very simple. Let us restrict to the case of hyperbolic manifolds of finite volume. Suppose that formula_18 is smaller than the Margulis constant for formula_45 and let formula_31 be a hyperbolic formula_21-manifold of finite volume. Then its thin part has two sorts of components:
In particular, a complete finite-volume hyperbolic manifold is always diffeomorphic to the interior of a compact manifold (possibly with empty boundary).
Other applications.
The Margulis lemma is an important tool in the study of manifolds of negative curvature. Besides the thick-thin decomposition some other applications are: