Complexity and T-invariant of Abelian and Milnor groups, and complexity of 3-manifolds

Abstract

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.