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

Thick and thin part of a manifold of finite volume

For a Riemannian manifold and a constant , the part is called the -thin part of the manifold ${\displaystyle M}$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon }$

${\displaystyle M_{<\epsilon }:=\left\{x\in M\colon \operatorname {inj} (x)<\epsilon \right\}}$

(where the injectivity radius is denoted in the point ) and the -thick part is the complement of the -thin part. ${\displaystyle \operatorname {inj} (x)}$ ${\displaystyle x}$ ${\displaystyle \epsilon }$ ${\displaystyle M_{\geq \epsilon }}$ ${\displaystyle \epsilon }$

${\displaystyle M_{<\epsilon }}$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. ${\displaystyle x\in M}$ ${\displaystyle \gamma \colon \left[0,1\right]\to M}$ ${\displaystyle l(\gamma )<2\epsilon }$ ${\displaystyle \gamma (0)=\gamma (1)=x}$ ${\displaystyle n}$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon _{n}}$

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 . ${\displaystyle n}$ ${\displaystyle \epsilon _{n}}$ ${\displaystyle n}$ ${\displaystyle \left[-1,0\right]}$ ${\displaystyle \epsilon <\epsilon _{n}}$ ${\displaystyle \epsilon }$ ${\displaystyle M_{<\epsilon }}$ ${\displaystyle \pi _{1}(M_{<\epsilon },x_{0})}$ ${\displaystyle <v(n)}$ ${\displaystyle n}$ ${\displaystyle v(n)}$

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 : ${\displaystyle n}$ ${\displaystyle \epsilon _{n}}$ ${\displaystyle n}$ ${\displaystyle \left[-1,0\right]}$ ${\displaystyle \epsilon <\epsilon _{n}}$

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 ${\displaystyle {\widetilde {M}}}$ ${\displaystyle n}$ ${\displaystyle \left[-1,0\right]}$ ${\displaystyle \Gamma }$ ${\displaystyle {\widetilde {M}}}$ ${\displaystyle x\in {\widetilde {M}}}$

${\displaystyle \left\{\gamma \in \Gamma \colon d(\gamma (x),x)<\epsilon \right\}}$

produced subgroup almost nilpotent. ${\displaystyle \Gamma _{\epsilon }(x)\subset \operatorname {Isom} ({\widetilde {M}})}$

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. ${\displaystyle \mu _{N}}$ ${\displaystyle <\mu _{N}}$ ${\displaystyle \Gamma \subset GL(N,\mathbb {C} )}$ ${\displaystyle A,B\subset O^{+}(n,1)}$ ${\displaystyle \max \left\{\parallel A-Id\parallel ,\parallel B-Id\parallel \right\}<2-{\sqrt {3}}}$

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. ${\displaystyle \Gamma =\pi _{1}(M,x_{0})}$ ${\displaystyle p\colon {\widetilde {M}}\to M}$ ${\displaystyle {\tilde {x}}_{0}\in p^{-1}(x_{0})\subset {\widetilde {M}}}$ ${\displaystyle d({\tilde {x}}_{0},\gamma {\tilde {x}}_{0})}$ ${\displaystyle \gamma \in \pi _{1}(M,x_{0})}$

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: ${\displaystyle M}$ ${\displaystyle \epsilon }$ ${\displaystyle M_{\geq \epsilon }}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \Gamma _{\epsilon }}$ ${\displaystyle \epsilon <\epsilon _{n}}$

• ${\displaystyle \Gamma _{\epsilon }}$is a group of parabolic isometries with the same fixed point at infinity
• ${\displaystyle \Gamma _{\epsilon }=\mathbb {Z} }$generated by a hyperbolic isometry
• ${\displaystyle \Gamma _{\epsilon }=\left\{id\right\}}$

This results in the following options for the topology of the connected components of the -thin part : ${\displaystyle \epsilon }$ ${\displaystyle M_{<\epsilon }}$

• ${\displaystyle V\times \left[0,\infty \right]}$for a closed flat manifold of dimension ${\displaystyle V}$ ${\displaystyle n-1}$
• ${\displaystyle B^{n-1}\times S^{1}}$ or ${\displaystyle S^{1}}$

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 ). ${\displaystyle \epsilon }$

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 . ${\displaystyle M}$ ${\displaystyle \mu (M)}$ ${\displaystyle \epsilon <\mu (M)}$

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. ${\displaystyle \mu (M)\geq 0{,}104}$ ${\displaystyle \mu (M)\leq 0{,}29}$ ${\displaystyle \mu (M)\leq 0{,}616}$

For hyperbolic manifolds of dimension we have ${\displaystyle n}$

${\displaystyle \mu (M)\geq {\frac {2^{\nu +1}}{3^{\nu +1}\pi ^{\nu }}}{\frac {\Gamma ({\frac {\nu +2}{2}})^{2}}{\Gamma (\nu +2)}}}$

with . Conversely, there is the inequality, which goes back to Kapovich , with an explicitly determinable constant . ${\displaystyle \nu =\left[{\frac {n+1}{2}}\right]}$ ${\displaystyle \epsilon _{n}\leq {\frac {C_{n}}{\sqrt {n}}}}$ ${\displaystyle C_{n}}$

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. ${\displaystyle l}$ ${\displaystyle w}$ ${\displaystyle \cosh \left({\tfrac {w}{2}}\right)=\coth \left({\tfrac {l}{2}}\right)}$ ${\displaystyle \lim _{l\to 0}\coth \left({\tfrac {l}{2}}\right)=\infty }$

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

1. ^ Benedetti-Petronio, Theorem D.2.2
2. ^ Benedetti-Petronio, Theorem D.3.3
3. ^ Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39 (5): 1038-1056, 1987.
4. ↑ Peter Shalen. Topology and geometry in dimension three, 103-109, Contemp. Math., 560, Amer. Math. Soc., Providence, RI, 2011.
5. ↑ Ruth Kellerhals . On the structure of hyperbolic manifolds. Israel J. Math. 143 (2004), 361-379.
6. ↑ Mikhail Belolipetsky. Hyperbolic orbifolds of small volume. (Appears in the Proceedings of the ICM 2014) pdf
7. ↑ 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
8. ^ Burton Randol: Cylinders in Riemann surfaces. Comment. Math. Helv. 54 (1979) no. 1, 1-5.