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

Abstract

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 $\{{\tilde{G}}_{\omega },\omega \in \Omega =\{0,1,2{\}}^{\mathbb{N}}\}$ 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.