On Normalish Subgroups of the R. Thompson Groups

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 congruent to...) (sic) Z) (sic) Z) (sic) 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.