Margulis lemma

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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 a set of isometries of S. Suppose there is a point x in S such that

d(f \cdot x,x)<\epsilon

for all f in F. Assume further that the subgroup \Gamma generated by F is discrete in Isom(S). Then \Gamma is virtually nilpotent. More precisely, there exists a subgroup \Gamma_0 in \Gamma which is nilpotent of nilpotency class at most r and of index at most N in \Gamma, where r and N are constants depending on S only.

The constant \epsilon(S) is often referred as the Margulis constant.


  • Werner Ballman, Mikhael Gromov, Victor Schroeder, Manifolds of Non-positive Curvature, Birkhauser, Boston (1985) p. 107