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On normalish subgroups of the R. Thompson groups

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Date
05/2020
Author
Bleak, Collin Patrick
Funder
EPSRC
Grant ID
EP/R032866/1
Keywords
Thompson's group
Amenable
C*-simplicity
Regular language
Synchronizing automata
Group actions
Normalish sub-groups
Wreath product
QA75 Electronic computers. Computer science
Theoretical Computer Science
Computer Science(all)
T-NDAS
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Abstract
Results in C∗ algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups F ≤ T ≤ V. These results together show that F is non-amenable if and only if T has a simple reduced C∗-algebra. In further investigations into the structure of C∗-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group G. They show that if a group G admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced C∗-algebra. Our chief result concerns the R. Thompson groups F < T < V; we show that there is an elementary amenable group E < F (where here, E ≅ ...)≀Z)≀Z)≀Z) with E normalish in V. The proof given uses a natural partial action of the group V on a regular language determined by a synchronizing automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of V with various forms of formal language theory.
Citation
Bleak , C P 2020 , On normalish subgroups of the R. Thompson groups . in N Jonoska & D Savchuk (eds) , Developments in Language Theory : 24th International Conference, DLT 2020, Tampa, FL, USA, May 11–15, 2020, Proceedings . Lecture Notes in Computer Science , vol. 12086 , Springer , pp. 29-42 , 24th International Conference on Developments in Language Theory (DLT) , Tampa , Florida , United States , 11/05/20 . https://doi.org/10.1007/978-3-030-48516-0_3
 
conference
 
Publication
Developments in Language Theory
DOI
https://doi.org/10.1007/978-3-030-48516-0_3
ISSN
0302-9743
Type
Conference item
Rights
© Springer Nature Switzerland AG 2020. 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://doi.org/10.1007/978-3-030-48516-0_3
Description
Funding: UK EPSRC grant EP/R032866/1
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  • University of St Andrews Research
URL
https://doi.org/10.1007/978-3-030-48516-0
URI
http://hdl.handle.net/10023/20020

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