Foundations of geometric approximate group theory

Cordes, Matthew; Hartnick, Tobias; Tonić, Vera

doi:10.48550/arxiv.2012.15303

Cited by 4 publications

(4 citation statements)

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“…Before we introduce the Lebesgue number with respect to subsets (i.e., subspaces) of a metric space, we need some additional terminology, which was also used in [CHT20].…”

Section: The Relative Lebesgue Number For a Covering Family Of A Subsetmentioning

confidence: 99%

The (largest) Lebesgue number and its relative version

Tonić

2023

Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znano

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In this paper we compare different definitions of the (largest) Lebesgue number of a cover U for a metric space X. We also introduce the relative version for the Lebesgue number of a covering family U for a subset A ⊆ X, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from [BL07], involving λ-quasi homothetic maps with coefficient R between metric spaces and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.

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“…Before we introduce the Lebesgue number with respect to subsets (i.e., subspaces) of a metric space, we need some additional terminology, which was also used in [CHT20].…”

Section: The Relative Lebesgue Number For a Covering Family Of A Subsetmentioning

confidence: 99%

The (largest) Lebesgue number and its relative version

Tonić

2023

Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znano

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“…Remark 5.3.7. We refer to the recent monograph [CHT20] for interesting examples of approximate subgroups of various set-groups. We should remark that their study of "geometric approximate group theory" is only tangentially related with our notion of coarse groups.…”

Section: 𝜾 𝒊 𝜾mentioning

confidence: 99%

An Invitation to Coarse Groups

Leitner¹,

Vigolo²

2022

Preprint

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In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to uniformly bounded error". In the first part of this work, we develop the theory of coarse homomorphisms, quotients, and subgroups, and prove that coarse versions of the Isomorphism Theorems hold true. We also initiate the study of coarse actions and show how they relate to the fundamental observation of Geometric Group Theory.In the second part we explore a selection of specialized topics, such as the study of coarse group structures on set-groups, groups of coarse automorphisms and spaces of controlled maps. Here the main aim is to show how the theory of coarse groups connects with classical subjects. These include: number theory; the study of bi-invariant metrics on groups; quasimorphisms and stable commutator length; Out(𝐹 𝑛 ); topological group actions.We see this text as an invitation and a stepping stone for further research.

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“…Theorem 2.1.30 (Polynomial growth theorem for approximate subgroups, Appendix, [CHT20]). Let Λ be an approximate subgroup of some group.…”

Section: A Primer On Infinite Approximate Subgroupsmentioning

confidence: 99%