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In [[mathematics]], the '''Margulis lemma''' (named after [[Grigory Margulis]]) is a result about [[discrete subgroup]]s of isometries of a [[symmetric space]] (e.g. the [[hyperbolic space|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 | |||
:<math>\epsilon=\epsilon(S)>0</math> | |||
with the following property. Let F be subset of isometries of '''S'''. Suppose there is a point x in '''S''' such that | |||
:<math>d(f \cdot x,x)<\epsilon</math> | |||
for all f in F. Assume further that the subgroup <math>\Gamma</math> generated by F is discrete in Isom('''S'''). Then <math>\Gamma</math> is virtually nilpotent. More precisely, there exists a subgroup <math>\Gamma_0</math> in <math>\Gamma</math> which is [[Nilpotent group|nilpotent]] of nilpotency class at most r and of [[Index of a subgroup|index]] at most N in <math>\Gamma</math>, where r and N are constants depending on '''S''' only. | |||
The constant <math>\epsilon(S)</math> is often referred as the ''Margulis constant''. | |||
==References== | |||
{{refbegin}} | |||
*Werner Ballman, Mikhael Gromov, Victor Schroeder, ''Manifolds of Non-positive Curvature'', Birkhauser, Boston (1985) p. 107 | |||
{{refend}} | |||
[[Category:Hyperbolic geometry]] | |||
[[Category:Differential geometry]] | |||
[[Category:Lemmas]] | |||
Latest revision as of 07:48, 16 August 2013
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
with the following property. Let F be subset of isometries of S. Suppose there is a point x in S such that
for all f in F. Assume further that the subgroup generated by F is discrete in Isom(S). Then is virtually nilpotent. More precisely, there exists a subgroup in which is nilpotent of nilpotency class at most r and of index at most N in , where r and N are constants depending on S only.
The constant is often referred as the Margulis constant.
References
- Werner Ballman, Mikhael Gromov, Victor Schroeder, Manifolds of Non-positive Curvature, Birkhauser, Boston (1985) p. 107