Generalized Grigorchuk’s overgroups as points in the space of marked 8-generated groups

Samarakoon, Supun T.

doi:10.1142/s021949882250058x

Cited by 1 publication

(1 citation statement)

References 12 publications

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“…For instance the closure

consists of a Cantor set

and a countable set

of isolated points accumulating to

and consisting of virtually metabelian groups containing a direct product of copies of the Lamplighter group

[ 15 , 78 ] as a subgroup of finite index. The closure of

is described in [ 79 ] and has a more complicated structure. More on spaces

and

see [ 18 , 80 , 81 ].…”

Section: Space Of Groups and Graphs And Approximationmentioning

confidence: 99%

Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

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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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