Quasi‐isometric diversity of marked groups

Minasyan, Ashot; Osin, Denis; Witzel, Stefan

doi:10.1112/topo.12187

Cited by 5 publications

(5 citation statements)

References 35 publications

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“…Note that some results of [Cha00] were revisited and improved on in [MOW19]. Since finite groups are hyperbolic, we have the inclusion F ⊆ H. Note that it is still an open question to prove that this inclusion is strict, equivalent to the problem of finding a non residually finite hyperbolic group.…”

Section: 3mentioning

confidence: 99%

“…The topology on the space of marked groups can be traced back to Chabauty in [Cha50]. It has by now become a standard tool in the study of group properties, see for instance [MOW19] and [Osi21] where tools of descriptive set theory are used to obtain far reaching group theoretical results. We will still include a brief introduction to describe the topology and the metric of the space of marked groups, which we denote by G throughout.…”

Section: Introductionmentioning

confidence: 99%

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Computable analysis on the space of marked groups

Rauzy¹

2021

Preprint

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We investigate decision problems for groups described by word problem algorithms. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. However, we prove that the space of marked groups is a Polish space that is not effectively Polish, because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and different theorems of Moschovakis. The space of marked groups will thus be an interesting object of study from the point of view of computable analysis alone, providing a natural space on which to test results that could apply to spaces that are not effectively Polish.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computable analysis on the space of marked groups

Rauzy¹

2021

Preprint

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“…While we have used the framework of Bowditch's taut loop length spectrum it is also possible to use the work of [12]. Let G be the space of marked groups.…”

Section: Uncountably Many Quasi-isometry Classesmentioning

confidence: 99%

“…Combining Lemma 4.3 and Lemma 7.1 we see that Z → G given by σ → G L (σ) is a continuous injection with perfect image. Thus, by [12,Theorem 1.1] we obtain uncountably many quasi-isometry classes of groups of the form G L (σ). By carefully choosing subsets of Z to ensure that the image is still perfect one can proves analogues of Theorem 7.5 for other properties satisfied by the G L (σ).…”

Section: Uncountably Many Quasi-isometry Classesmentioning

confidence: 99%

Constructing groups of type $FP_2$ over fields but not over the integers

Kropholler¹

2021

Preprint

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“…The closure of { G ω } ω∈Ω is described in [74] and has a more complicated structure. More on spaces M m and M * Sch (m) see [18,75,76].…”

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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Quasi‐isometric diversity of marked groups

Cited by 5 publications

References 35 publications

Computable analysis on the space of marked groups

Computable analysis on the space of marked groups

Constructing groups of type $FP_2$ over fields but not over the integers

Integrable and Chaotic Systems Associated with Fractal Groups

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