On Growth of Generalized Grigorchuk's Overgroups

Abstract

Grigorchuk's Overgroup $\tilde{\mathcal{G}}$, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group $\mathcal{G}$ of intermediate growth constructed in 1980, but also has elements of infinite order. It's growth is substantially greater than the growth of $\mathcal{G}$. The group $\mathcal{G}$, corresponding to the sequence $(012{)}^{\infty }=\mathrm{012012...}$, is a member of the family $\{G_{\omega }|\omega \in \Omega =\{0,1,2{\}}^{\mathbb{N}}\}$ consisting of groups of intermediate growth when sequence $\omega$ is not virtually constant. Following this construction we define the family $\{{\tilde{G}}_{\omega },\omega \in \Omega \}$ of generalized overgroups. Then $\tilde{\mathcal{G}}={\tilde{G}}_{(012{)}^{\infty }}$ and $G_{\omega }$ is a subgroup of ${\tilde{G}}_{\omega }$ for each $\omega \in \Omega$. We prove, if $\omega$ is eventually constant, then ${\tilde{G}}_{\omega }$ is of polynomial growth and if $\omega$ is not eventually constant, then ${\tilde{G}}_{\omega }$ is of intermediate growth.

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