Lemma of Margulis
In differential geometry , a branch of mathematics , the lemma of Margulis or Margulis lemma describes the topology of the “thin part” of a negatively curved Riemann manifold. It is mainly used to describe the ends of hyperbolic manifolds of finite volumes.
It is named after Grigory Alexandrovich Margulis .
Thin and thick part of a manifold
For a Riemannian manifold and a constant , the part is called the -thin part of the manifold
(where the injectivity radius is denoted in the point ) and the -thick part is the complement of the -thin part.
is therefore that the amount to which there is a closed , not nullhomotope , curve the length to exist. Frequently, it is called only from the thin and thick part of a - dimensional manifold , referring to -dünnen and thicknesses for a part which is smaller than the defined below Margulis constant is.
Lemma of Margulis (differential geometric formulation)
Margulis' lemma says that there is a Margulis constant for every dimension , so that for all complete Riemannian manifolds of the dimension with intersecting curvatures in the interval and for all of the -thin part, there is almost nilpotent fundamental group : there is a nilpotent subgroup of Index for a constant that only depends on .
Lemma of Margulis (group theory formulation)
Margulis' lemma says that there is a Margulis constant for every dimension , so that the following statement applies to all simply connected complete Riemannian manifolds of the dimension with intersecting curvatures in the interval and for all :
Let it be a simply connected Riemannian manifold of dimension with intersection curvatures in the interval . Let be an actually discontinuous group of isometries of and . Then that of
produced subgroup almost nilpotent.
The special case for matrix groups is also known as the Zassenhaus lemma : There is a constant such that every discrete subgroup generated by matrices of the norm is almost nilpotent. In fact, the following elementary lemma , which goes back to Hans Zassenhaus , holds : If two matrices create a discrete group and hold, then A and B commute.
The group theoretical and differential geometric formulation of Margulis lemma are equivalent to each other. The equivalence is obtained by means of the action of on the universal overlay . For an archetype corresponds to the length of the shortest representative closed path.
Ends of hyperbolic manifolds of finite volume
Let it be a complete hyperbolic manifold of finite volume . Then the thick part is compact (for any ) and for the group there are (for ) the following options:
- is a group of parabolic isometries with the same fixed point at infinity
- generated by a hyperbolic isometry
This results in the following options for the topology of the connected components of the -thin part :
- for a closed flat manifold of dimension
- or
In the first case, there are so-called tips ( cusps ). In the second case, it is about tube environments of closed geodesics (or closed geodesics of length ).
Margulis numbers
For a Riemannian manifold , the Margulis number is the greatest real number, so that the conclusion of the Margulis lemma holds for all .
For hyperbolic 3-manifolds is . Peter Shalen proved that holds for almost all hyperbolic 3-manifolds . Based on numerical calculations, it is assumed that always applies.
For hyperbolic manifolds of dimension we have
with . Conversely, there is the inequality, which goes back to Kapovich , with an explicitly determinable constant .
Collar lemma
From Margulis' lemma it can be deduced that very short closed geodesics must have a collar area of large hyperbolic volume. A quantitative description of this relationship for areas provides the Collar Lemma (ger .: collar lemma ), the first version in 1974 by Linda Keen was proved. The best possible estimate goes back to Randol: in a hyperbolic surface having a closed geodesic the length of a collar around the width with . Note that is.
literature
- Každan, DA ; Margulis, GA: A proof of Selberg's hypothesis. (Russian) Mat. Sb. (NS) 75 (117) 1968, 163-168.
- Raghunathan, MS : Discrete subgroups of Lie groups. Results of Mathematics and its Frontier Areas, Volume 68. Springer-Verlag, New York-Heidelberg, 1972.
- Buser, Peter ; Karcher, Hermann : Gromov's almost flat manifolds. Astérisque, 81st Société Mathématique de France, Paris, 1981.
- Ballmann, Werner ; Gromov, Mikhael ; Schroeder, Viktor: Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkhauser Boston, Inc., Boston, MA, 1985. ISBN 0-8176-3181-X
- Benedetti, Riccardo; Petronio, Carlo: Lectures on hyperbolic geometry. University text. Springer-Verlag, Berlin, 1992. ISBN 3-540-55534-X
Web links
- Bromberg: The Thick-Thin Decomposition
- Gallot: Margulis Lemmas without curvature
- Shalen: Margulis Numbers of Hyperbolic 3-Manifolds
- Bergeron, Guilloux: Géométrie hyperbolique et représentations de groupes de surface (Chapitre III: Théorème de Bieberbach et lemme de Margulis)
- Martelli: Geometric Topology (Chapter 4: Thin-thick decomposition)
Individual evidence
- ^ Benedetti-Petronio, Theorem D.2.2
- ^ Benedetti-Petronio, Theorem D.3.3
- ^ Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39 (5): 1038-1056, 1987.
- ↑ Peter Shalen. Topology and geometry in dimension three, 103-109, Contemp. Math., 560, Amer. Math. Soc., Providence, RI, 2011.
- ↑ Ruth Kellerhals . On the structure of hyperbolic manifolds. Israel J. Math. 143 (2004), 361-379.
- ↑ Mikhail Belolipetsky. Hyperbolic orbifolds of small volume. (Appears in the Proceedings of the ICM 2014) pdf
- ↑ Linda Keen: Collars on Riemann surfaces. Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 263-268. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, NJ, 1974
- ^ Burton Randol: Cylinders in Riemann surfaces. Comment. Math. Helv. 54 (1979) no. 1, 1-5.