Characterization of finitely generated groups by types

Abstract

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.