Polyproline helix

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

$ϵ=ϵ(S)>0$

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

$d(f\cdot x,x)<ϵ$

for all f in F. Assume further that the subgroup $\mathrm{\Gamma }$ generated by F is discrete in Isom(S). Then $\mathrm{\Gamma }$ is virtually nilpotent. More precisely, there exists a subgroup ${\mathrm{\Gamma }}_0$ in $\mathrm{\Gamma }$ which is nilpotent of nilpotency class at most r and of index at most N in $\mathrm{\Gamma }$, where r and N are constants depending on S only.

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

References

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