On growth of generalized Grigorchuk's overgroups

Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.