Margulis lemma

Formal statement The Margulis lemma can be formulated as follows. Let X be a simply-connected manifold of non-positive bounded sectional curvature. There exist constants C, \varepsilon>0 with the following property. For any discrete subgroup \Gamma of the group of isometries of X and any x \in X, if F_x is the set: : F_x = \{ g \in \Gamma: d(x, gx) then the subgroup generated by F_x contains a nilpotent subgroup of index less than C. Here d is the distance induced by the Riemannian metric. An immediately equivalent statement can be given as follows: for any subset F of the isometry group, if it satisfies that: • there exists a x \in X such that \forall g \in F : d(x, gx) ; • the group \langle F \rangle generated by F is discrete then \langle F \rangle contains a nilpotent subgroup of index \le C. Margulis constants The optimal constant \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 2 \operatorname{arsinh} \left( \sqrt{\frac {2\cos(2\pi/7) - 1} {8\cos(\pi/7) + 7}} \right) \simeq 0.2629 ; • In general the Margulis constant \varepsilon_n for the hyperbolic n-space is known to satisfy the bounds: c^{-n^2} for some 0 0. Zassenhaus neighbourhoods 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. :If G is a semisimple Lie group there exists a neighbourhood \Omega of the identity in G and a C > 0 such that any discrete subgroup \Gamma which is generated by \Gamma \cap \Omega contains a nilpotent subgroup of index \le C. Such a neighbourhood \Omega is called a Zassenhaus neighbourhood in G. If G is compact this theorem amounts to Jordan's theorem on finite linear groups. == Thick-thin decomposition ==

Let M be a Riemannian manifold and \varepsilon > 0. The thin part of M is the subset of points x \in M where the injectivity radius of M at x is less than \varepsilon, usually denoted M_{, and the thick part its complement, usually denoted M_{\ge \varepsilon}. There is a tautological decomposition into a disjoint union M = M_{. When M is of negative curvature and \varepsilon is smaller than the Margulis constant for the universal cover \widetilde M, the structure of the components of the thin part is very simple. Let us restrict to the case of hyperbolic manifolds of finite volume. Suppose that \varepsilon is smaller than the Margulis constant for \mathbb H^n and let M be a hyperbolic n-manifold of finite volume. Then its thin part has two sorts of components: • Cusps: these are the unbounded components, they are diffeomorphic to a flat (n-1)-manifold times a line; • Margulis tubes: these are neighbourhoods of closed geodesics of length on M. They are bounded and (if M is orientable) diffeomorphic to a circle times a (n-1)-disc. In particular, a complete finite-volume hyperbolic manifold is always diffeomorphic to the interior of a compact manifold (possibly with empty boundary). == Other applications ==

The Margulis lemma is an important tool in the study of manifolds of negative curvature. Besides the thick-thin decomposition some other applications are: • The collar lemma: this is a more precise version of the description of the compact components of the thin parts. It states that any closed geodesic of length \ell on an hyperbolic surface is contained in an embedded cylinder of diameter of order \ell^{-1}. • The Margulis lemma gives an immediate qualitative solution to the problem of minimal covolume among hyperbolic manifolds: since the volume of a Margulis tube can be seen to be bounded below by a constant depending only on the dimension, it follows that there exists a positive infimum to the volumes of hyperbolic n-manifolds for any n. • The existence of Zassenhaus neighbourhoods is a key ingredient in the proof of the Kazhdan–Margulis theorem. • One can recover the Jordan–Schur theorem as a corollary to the existence of Zassenhaus neighbourhoods. ==See also==