Margulis lemma

The Margulis lemma can be formulated as follows.[1]

Let ${\displaystyle X}$ be a simply-connected manifold of non-positive bounded sectional curvature. There exist constants ${\displaystyle C,\varepsilon >0}$ with the following property. For any discrete subgroup ${\displaystyle \Gamma }$ of the group of isometries of ${\displaystyle X}$ and any ${\displaystyle x\in X}$, if ${\displaystyle F_{x}}$ is the set:

${\displaystyle F_{x}=\{g\in \Gamma :d(x,gx)<\varepsilon \}}$

then the subgroup generated by ${\displaystyle F_{x}}$ contains a nilpotent subgroup of index less than ${\displaystyle C}$. Here ${\displaystyle d}$ is the distance induced by the Riemannian metric.

An immediately equivalent statement can be given as follows: for any subset ${\displaystyle F}$ of the isometry group, if it satisfies that:

• there exists a ${\displaystyle x\in X}$ such that ${\displaystyle \forall g\in F:d(x,gx)<\varepsilon }$;
• the group ${\displaystyle \langle F\rangle }$ generated by ${\displaystyle F}$ is discrete

then ${\displaystyle \langle F\rangle }$ contains a nilpotent subgroup of index ${\displaystyle \leq C}$.

The optimal constant ${\displaystyle \varepsilon }$ 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 ${\displaystyle 2\operatorname {arsinh} \left({\sqrt {\frac {2\cos(2\pi /7)-1}{8\cos(\pi /7)+7}}}\right)\simeq 0.2629}$;[2]
• In general the Margulis constant ${\displaystyle \varepsilon _{n}}$ for the hyperbolic ${\displaystyle n}$-space is known to satisfy the bounds:
  $${\displaystyle c^{-n^{2}}<\varepsilon _{n}<K/{\sqrt {n}}}$$

  
  for some ${\displaystyle 0<c<1,K>0}$.[3]

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.[4]

If ${\displaystyle G}$ is a semisimple Lie group there exists a neighbourhood ${\displaystyle \Omega }$ of the identity in ${\displaystyle G}$ and a ${\displaystyle C>0}$ such that any discrete subgroup ${\displaystyle \Gamma }$ which is generated by ${\displaystyle \Gamma \cap \Omega }$ contains a nilpotent subgroup of index ${\displaystyle \leq C}$.

Such a neighbourhood ${\displaystyle \Omega }$ is called a Zassenhaus neighbourhood in ${\displaystyle G}$. If ${\displaystyle G}$ is compact this theorem amounts to Jordan's theorem on finite linear groups.