The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r

Collin Bleak; Peter Cameron; Yonah Maissel; Andrés Navas; Feyishayo Olukoya

doi:10.1090/memo/1510

The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G_n,r

Collin Bleak, Peter Cameron, Yonah Maissel, Andrés Navas, Feyishayo Olukoya

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

Abstract

We characterise the automorphism groups of the Higman groups G_n,r as groups of specific homeomorphisms of Cantor spaces C_n,r, through the use of Rubin's theorem. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the 'Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of V, which was the 'last chameleon.' As it happens, the homeomorphisms which arise naturally fit into the framework of Grigorchuk, Nekrashevich, and Suschanskiī's rational group of transducers, and exhibit fascinating connections with the theory of reset words for automata (arising in the Road Colouring Problem), while also appearing to offer insight into the nature of Brin and Guzmán's exotic automorphisms.

Original language	English
Number of pages	44
Journal	Memoirs of the American Mathematical Society
Volume	301
Issue number	1510
DOIs	https://doi.org/10.1090/memo/1510
Publication status	Published - 23 Sept 2024

Keywords

Automorphism groups
Higman-Thompson groups
Chameleon groups
Rational group
Transducers

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10.1090/memo/1510

Automorphisms_via_dynamics_memoirs_revised
Copyright © 2022 the Author(s). This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo.
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https://arxiv.org/abs/1605.09302

Cite this

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title = "The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r",

abstract = "We characterise the automorphism groups of the Higman groups Gn,r as groups of specific homeomorphisms of Cantor spaces Cn,r, through the use of Rubin's theorem. This continues a thread of research begun by Brin, and extended later by Brin and Guzm{\'a}n: to characterise the automorphism groups of the 'Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of V, which was the 'last chameleon.' As it happens, the homeomorphisms which arise naturally fit into the framework of Grigorchuk, Nekrashevich, and Suschanskiī's rational group of transducers, and exhibit fascinating connections with the theory of reset words for automata (arising in the Road Colouring Problem), while also appearing to offer insight into the nature of Brin and Guzm{\'a}n's exotic automorphisms.",

keywords = "Automorphism groups, Higman-Thompson groups, Chameleon groups, Rational group, Transducers",

author = "Collin Bleak and Peter Cameron and Yonah Maissel and Andr{\'e}s Navas and Feyishayo Olukoya",

note = "Funding: The first and second authors wish to acknowledge support from EPSRC grant EP/R032866/1 received during the editing process of this article. The fourth author would like to thank St. Andrews University for its hospitality during the Workshop on the Extended Family of Thompson{\textquoteright}s Groups in 2014, and acknowledges the support of DySYRF (Anillo Project 1103, CONICYT) and Fondecyt{\textquoteright}s project 1120131. The fifth author was partly supported by Leverhulme Trust Research Project Grant RPG-2017-159. ",

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AU - Navas, Andrés

AU - Olukoya, Feyishayo

N1 - Funding: The first and second authors wish to acknowledge support from EPSRC grant EP/R032866/1 received during the editing process of this article. The fourth author would like to thank St. Andrews University for its hospitality during the Workshop on the Extended Family of Thompson’s Groups in 2014, and acknowledges the support of DySYRF (Anillo Project 1103, CONICYT) and Fondecyt’s project 1120131. The fifth author was partly supported by Leverhulme Trust Research Project Grant RPG-2017-159.

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N2 - We characterise the automorphism groups of the Higman groups Gn,r as groups of specific homeomorphisms of Cantor spaces Cn,r, through the use of Rubin's theorem. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the 'Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of V, which was the 'last chameleon.' As it happens, the homeomorphisms which arise naturally fit into the framework of Grigorchuk, Nekrashevich, and Suschanskiī's rational group of transducers, and exhibit fascinating connections with the theory of reset words for automata (arising in the Road Colouring Problem), while also appearing to offer insight into the nature of Brin and Guzmán's exotic automorphisms.

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KW - Rational group

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The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G_n,r

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Keywords

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

Collin Patrick Bleak

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The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r

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

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Collin Patrick Bleak

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The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G_n,r