On growth of generalized Grigorchuk's overgroups

Abstract

Grigorchuk’s Overgroup Ĝ, 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 { Ĝω, ω ∈ Ω} of generalized overgroups. Then Ĝ = Ĝ (012)∞ and Gω is a subgroup of Ĝω for each ω ∈ Ω. We prove, if ω is eventually constant, then Ĝω is of polynomial growth and if ω is not eventually constant, then Ĝω is of intermediate growth.