On growth of generalized Grigorchuk's overgroups

Abstract

Grigorchuk's Overgroup \(\widetilde{\mathcal{G}}\), 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. Its growth is substantially greater than the growth of \(\mathcal{G}\). The group \(\mathcal{G}\), corresponding to the sequence \((012)^\infty = 012012 \dots\), 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 eventually constant. Following this construction we define the family \(\{ \widetilde{G}_\omega, \omega \in \Omega \}\) of generalized overgroups. Then \(\widetilde{\mathcal{G}} = \widetilde{G}_{(012)^\infty}\) and \(G_\omega\) is a subgroup of \(\widetilde{G}_\omega\) for each \(\omega \in \Omega\). We prove, if \(\omega\) is eventually constant, then \(\widetilde{G}_\omega\) is of polynomial growth and if \(\omega\) is not eventually constant, then \(\widetilde{G}_\omega\) is of intermediate growth.