Supun T. Samarakoon | Searcharxiv

arXiv2019

Generalized Grigorchuk's Overgroups as points on $\mathcal{M}_k$

Following the construction from `Degrees of growth of finitely generated groups and the theory of invariant means' we generalize the Grigorchuk's overgroup $\tilde{\mathcal{G}}$, studied in `On parabolic subgroups and Hecke algebras of some fractal groups' to the family $\{ \tilde{G}_ω, ω\in Ω= \{ 0, 1, 2 \}^\mathbb{N} \}$ of generalized Grigorchuk's overgroups. We consider these groups as 8-generated and describe the closure of this family in the space $\mathcal{M}_8$ of marked groups.

math.GR↗

arXiv2019

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 $\{ G_ω| ω\in Ω= \{ 0, 1, 2 \}^\mathbb{N} \}$ consisting of groups of intermediate growth when sequence $ω$ is not virtually constant. Following this construction we define the family $\{ \tilde{G}_ω, ω\in Ω\}$ of generalized overgroups. Then $\tilde{\mathcal{G}} = \tilde{G}_{(012)^\infty}$ and $G_ω$ is a subgroup of $\tilde{G}_ω$ for each $ω\in Ω$. We prove, if $ω$ is eventually constant, then $\tilde{G}_ω$ is of polynomial growth and if $ω$ is not eventually constant, then $\tilde{G}_ω$ is of intermediate growth.

math.GR↗