Subnormal subgroups in free groups, their growth and cogrowth

In this paper, the author (1) compare subnormal closures of finite sets in a free group F; (2) obtains the limit for the series of subnormal closures of a single element in F; (3) proves that the exponential growth rate (exp.g.r.) $\lim_{n\to \infty}\sqrt[n]{g_H(n)}$ , where gH (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; (4) gives sharp estimates from below for the exp.g.r. of subnormal subgroups in free groups; and (5) finds the cogrowth of the subnormal closures of free generators.