On Growth of Generalized Grigorchuk's Overgroups

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 $(012)^\infty = 012012 ...$, 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.