Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.

First Grigorchuk group [Formula: see text] and Grigorchuk’s overgroup [Formula: see text], introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, [Formula: see text] was used to construct the family of generalized Grigorchuk groups [Formula: see text], which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup [Formula: see text] to the family [Formula: see text] of generalized Grigorchuk’s overgroups. We consider these groups as 8-generated and describe the closure of this family in the space [Formula: see text] of marked [Formula: see text]-generated groups.

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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