On the growth of rank for subgroups of finitely generated groups

In [1] and [2] the functions of rank growth were independently introduced and investigated for subgroups of a finitely generated free group. In the present paper the concept of growth of rank is extended to subgroups of an arbitrary finitely generated group $G$, and the dependence of the asymptotic behaviour of the above functions on the choice of a finite generating set in $G$ is studied. For a broad class of groups (which includes, in particular, the free polynilpotent groups) estimates for the growth of rank for subgroups are obtained that generalize the wellknown Baumslag–Eidel'kind result on finitely generated normal subgroups. Some problems related to the realization of arbitrary functions as functions of rank growth for subgroups of soluble groups are treated.