Rational embeddings of hyperbolic groups

James Belk; Collin Bleak; Francesco Matucci

doi:10.4171/JCA/52

Rational embeddings of hyperbolic groups

James Belk^*, Collin Bleak, Francesco Matucci

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskii. The proof involves assigning a system of binary addresses to points in the Gromov boundary of a hyperbolic group G, and proving that elements of G act on these addresses by asynchronous transducers. These addresses derive from a certain self-similar tree of subsets of G, whose boundary is naturally homeomorphic to the horofunction boundary of G.

Original language	English
Pages (from-to)	123-183
Number of pages	61
Journal	Journal of Combinatorial Algebra
Volume	5
Issue number	2
DOIs	https://doi.org/10.4171/JCA/52
Publication status	Published - 15 Jun 2021

Keywords

Hyperbolic groups
Rational group
Gromov boundary
Horofunction boundary
Transducers

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10.4171/JCA/52Licence: CC BY

Belk-2021-Rational-embeddings-JCA-5-2-123-CCBY
Copyright © 2021 European Mathematical Society Published by EMS Press. This work is licensed under a CC BY 4.0 license.
Final published version, 2.02 MBLicence: CC BY

Bi-synchronizing automata: Bi-synchronizing automata, outer automorphism groups of Higman-Thompson groups, and automorphisms of the shift
Bleak, C. P. (PI) & Cameron, P. J. (CoI)
1/05/18 → 30/04/21
Project: Standard

Cite this

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title = "Rational embeddings of hyperbolic groups",

abstract = "We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskii. The proof involves assigning a system of binary addresses to points in the Gromov boundary of a hyperbolic group G, and proving that elements of G act on these addresses by asynchronous transducers. These addresses derive from a certain self-similar tree of subsets of G, whose boundary is naturally homeomorphic to the horofunction boundary of G.",

keywords = "Hyperbolic groups, Rational group, Gromov boundary, Horofunction boundary, Transducers",

author = "James Belk and Collin Bleak and Francesco Matucci",

note = "Funding: The first and second authors have been partially supported by EPSRC grant EP/R032866/1 during the creation of this paper. The third author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM) and gratefully acknowledges the support of the Funda{\c c}{\~a}o de Amparo {\`a} Pesquisa do Estado de S{\~a}o Paulo (FAPESP Jovens Pesquisadores em Centros Emergentes grant 2016/12196-5), of the Conselho Nacional de Desenvolvimento Cientf{\'i}co e Tecnol{\'o}gico (CNPq Bolsa de Produtividade emPesquisa PQ-2 grant 306614/2016-2) and of the Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia (CEMAT-Ci{\^e}ncias FCT projects UIDB/04621/2020 and UIDP/04621/2020).",

year = "2021",

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doi = "10.4171/JCA/52",

language = "English",

volume = "5",

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journal = "Journal of Combinatorial Algebra",

issn = "2415-6302",

publisher = "European Mathematical Society Publishing House",

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T1 - Rational embeddings of hyperbolic groups

AU - Belk, James

AU - Bleak, Collin

AU - Matucci, Francesco

N1 - Funding: The first and second authors have been partially supported by EPSRC grant EP/R032866/1 during the creation of this paper. The third author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM) and gratefully acknowledges the support of the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP Jovens Pesquisadores em Centros Emergentes grant 2016/12196-5), of the Conselho Nacional de Desenvolvimento Cientfíco e Tecnológico (CNPq Bolsa de Produtividade emPesquisa PQ-2 grant 306614/2016-2) and of the Fundação para a Ciência e a Tecnologia (CEMAT-Ciências FCT projects UIDB/04621/2020 and UIDP/04621/2020).

PY - 2021/6/15

Y1 - 2021/6/15

N2 - We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskii. The proof involves assigning a system of binary addresses to points in the Gromov boundary of a hyperbolic group G, and proving that elements of G act on these addresses by asynchronous transducers. These addresses derive from a certain self-similar tree of subsets of G, whose boundary is naturally homeomorphic to the horofunction boundary of G.

AB - We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskii. The proof involves assigning a system of binary addresses to points in the Gromov boundary of a hyperbolic group G, and proving that elements of G act on these addresses by asynchronous transducers. These addresses derive from a certain self-similar tree of subsets of G, whose boundary is naturally homeomorphic to the horofunction boundary of G.

KW - Hyperbolic groups

KW - Rational group

KW - Gromov boundary

KW - Horofunction boundary

KW - Transducers

U2 - 10.4171/JCA/52

DO - 10.4171/JCA/52

M3 - Article

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VL - 5

SP - 123

EP - 183

JO - Journal of Combinatorial Algebra

JF - Journal of Combinatorial Algebra

IS - 2

ER -

Rational embeddings of hyperbolic groups

Abstract

Keywords

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Projects

Bi-synchronizing automata: Bi-synchronizing automata, outer automorphism groups of Higman-Thompson groups, and automorphisms of the shift

Cite this