On Growth of Generalized Grigorchuk's Overgroups

Abstract

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. It's 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 virtually constant. Following this construction we define the family { Gω, ω ∈ Ω} of generalized overgroups. Then G = G(012) ∞ and Gω is a subgroup of Gω for each ω ∈ Ω. We prove, if ω is eventually constant, then Gω is of polynomial growth and if ω is not eventually constant, then Gω is of intermediate growth.