  In this paper, the author (1) compares subnormal closures of finite sets in
free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the
limit of the n-th roots of g(n), where g(n) is the growth function of a
subgroup H with respect to a finite free basis of F, exists for any subgroup H
of the free group F; (3) gives sharp estimates from below for the e.g.r. of
subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal
closures of free generators in F.
