I have read sometimes conclusions pertaining nilpotent groups in geometrical settings. Particularly I read that Fukaya and Yamaguchi proved that if $M_i$ is a sequence of Riemanian manifolds converging to a single-point set (in the Gromov-Hasusdorff topology), then $\pi_1(M_i)$ has a nilpotent subgroup of finite index.
In general terms, why is this important? What does it say about the geometry (or topology) of $M_i$? I don't really have a deep understanding of nilpotent groups, other than the definition and I would like to know what is the geometrical meaning of results concerning these.