  We investigate the notion of complexity for finitely presented groups and the
related notion of complexity for three-dimensional manifolds. We give two-sided
estimates on the complexity of all the Milnor groups (the finite groups with
free action on the three-sphere), as well as for all finite Abelian groups. The
ideas developed in the process also allow to construct two-sided bounds for the
values of the so-called T-invariant (introduced by Delzant) for the above
groups, and to estimate from below the value of T-invariant for an arbitrary
finitely presented group. Using the results of this paper and of previous ones,
we then describe an infinite collection of Seifert three-manifolds for which we
can asymptotically determine the complexity in an exact fashion up to linear
functions. We also provide similar estimates for the complexity of several
infinite families of Milnor groups.
