On Growth of Generalized Grigorchuk's Overgroups

2019/09/03 by Samarakoon, Supun T.
#20E08 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1909.01272

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 \ G_ω| 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 \ G_ω, ω∈ Ω\ of generalized overgroups. Then G = G_(012)^∞ and G_ω 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.

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