Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS 47
Theorem. Given C and n there exists m such that the following holds: Let ε > 0,
and let Γ be a group containing a nilpotent subgroup N of index ≤ C which has a
nilpotent basis of length ≤ n. Then there is a compact m-dimensional manifold M
with sectional curvature K > −1 and a point p ∈ M such that Γ is isomorphic to
the image of the homomorphism
π 1
(
Bε (p), p ) → π 1 (M, p).
Apart from the issue of finding the optimal dimension another difference to
Theorem 1 is that this theorem uses the homomorphism to π 1 (M) rather than to
π 1 (B 1 (p), p). This actually allows for more flexibility (by adding relations to the
fundamental group at large distances to p). This is the reason why the following
problem remains open.
Problem. The most important problem in the context of the Margulis Lemma
for manifolds with lower Ricci curvature bound that remains open is whether or
whether not one can arrange in Theorem 1 for the torsion of N to be abelian. We
refer the reader to [KPT10] for some related conjectures for manifolds with almost
nonnegative sectional curvature.
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