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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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Automorphisms_via_dynamics_memoirs_revised.pdf (731.7Kb)
Date
22/04/2022
Author
Bleak, Collin
Cameron, Peter
Maissel, Yonah
Navas, Andrés
Olukoya, Feyishayo
Funder
EPSRC
Grant ID
EP/R032866/1
Keywords
Automorphism groups
Higman-Thompson groups
Chameleon groups
Rational group
Transducers
QA Mathematics
T-NDAS
BDQ
AC
MCP
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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 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.
Citation
Bleak , C , Cameron , P , Maissel , Y , Navas , A & Olukoya , F 2022 , ' The further chameleon groups of Richard Thompson and Graham Higman : automorphisms via dynamics for the Higman groups G n,r ' , Memoirs of the American Mathematical Society . < https://arxiv.org/abs/1605.09302 >
Publication
Memoirs of the American Mathematical Society
Status
Peer reviewed
ISSN
0065-9266
Type
Journal article
Rights
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.
Description
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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  • University of St Andrews Research
URL
https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo
https://arxiv.org/abs/1605.09302
URI
http://hdl.handle.net/10023/26680

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