Margulis lemma

In mathematics, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a symmetric space (e.g. the hyperbolic n-space), or more generally a space of non-positive curvature.

Theorem: Let S be a Riemannian symmetric space of non-compact type. There is a positive constant

\epsilon=\epsilon(S)>0

with the following property. Let F be a set of isometries of S. Suppose there is a point x in S such that

d(f \cdot x,x)<\epsilon

for all f in F. Assume further that the subgroup $\Gamma$ generated by F is discrete in Isom(S). Then $\Gamma$ is virtually nilpotent. More precisely, there exists a subgroup $\Gamma_0$ in $\Gamma$ which is nilpotent of nilpotency class at most r and of index at most N in $\Gamma$ , where r and N are constants depending on S only.

The constant $\epsilon(S)$ is often referred as the Margulis constant.

References

Werner Ballman, Mikhael Gromov, Victor Schroeder, Manifolds of Non-positive Curvature, Birkhauser, Boston (1985) p. 107