Margulis lemma

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. Let be a simply-connected manifold of non-positive bounded sectional curvature. There exist constants with the following property. For any discrete subgroup of the group of isometries of and any , if is the set: then the subgroup generated by contains a nilpotent subgroup of index less than . Here 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: there exists a such that ; the group generated by is discrete then contains a nilpotent subgroup of index . The optimal constant 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: the optimal constant for the hyperbolic plane is equal to ; In general the Margulis constant for the hyperbolic -space is known to satisfy the bounds: for some . 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.