In differential geometry, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (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 can be formulated as follows.
be a simply-connected manifold of non-positive bounded sectional curvature.
is the distance induced by the Riemannian metric.
An immediately equivalent statement can be given as follows: for any subset
of the isometry group, if it satisfies that: then
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: 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.
is compact this theorem amounts to Jordan's theorem on finite linear groups.
, and the thick part its complement, usually denoted
There is a tautological decomposition into a disjoint union
is smaller than the Margulis constant for the universal cover
, the structure of the components of the thin part is very simple.
Let us restrict to the case of hyperbolic manifolds of finite volume.
is smaller than the Margulis constant for
Then its thin part has two sorts of components:[5] In particular, a complete finite-volume hyperbolic manifold is always diffeomorphic to the interior of a compact manifold (possibly with empty boundary).
The Margulis lemma is an important tool in the study of manifolds of negative curvature.