Margulis lemma

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In mathematics, the Margulis lemma 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

ε = ε(S) > 0

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 Γ generated by F is discrete in Isom(S). Then Γ is virtually nilpotent. More precisely, there exists a subgroup Γ₀ in Γ which is nilpotent of nilpotency class at most r and of index at most N in Γ, where r and N are constants depending on S only.

The constant ε(S) > 0 is often referred as the Margulis constant.

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