Normalish amenable subgroups of the R. Thompson groups

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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 synchronising 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.
Original languageEnglish
Pages (from-to)785-800
JournalInternational Journal of Foundations of Computer Science
Volume32
Issue number06
DOIs
Publication statusPublished - 9 Sept 2021

Keywords

  • Thompson's groups
  • Amenable
  • C*-simplicity
  • Regular language
  • Normalish

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