von Neumann Algebras of Thompson-like Groups from Cloning Systems II

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Publication date: 4 March 2023

Abstract: Let

(G_{n})_{n i n m a t h b b N}

be a sequence of groups equipped with a

d

-ary cloning system and denote by

m a t h s c r T_{d} (G_{*})

the resulting Thompson-like group. In previous work joint with Zaremsky, we obtained structural results concerning the group von Neumann algebra of

m a t h s c r T_{d} (G_{*})

, denoted by

L (m a t h s c r T_{d} (G_{*}))

. Under some natural assumptions on the

d

-ary cloning system, we proved that

L (m a t h s c r T_{d} (G_{*}))

is a type

e x t {I I}_{1}

factor. With a few additional natural assumptions, we proved that

L (m a t h s c r T_{d} (G_{*}))

is, moreover, a McDuff factor. In this paper, we further analyze the structure of

L (m a t h s c r T_{d} (G_{*}))

, in particular the inclusion

L (F_{d}) s u b s e t e q L (m a t h s c r T_{d} (G_{*}))

, where

F_{d}

is the smallest of the Higman--Thompson groups. First, we prove that if the

d

-ary cloning system is "diverse", then the inclusion

L (F_{d}) s u b s e t e q L (m a t h s c r T_{d} (G_{*}))

is irreducible, a considerable improvement of the result that

L (m a t h s c r T_{d} (G_{*}))

is a type

e x t {I I}_{1}

factor. This allows us to compute the normalizer of

L (F_{d})

in

L (m a t h s c r T_{d} (G_{*}))

, which turns out to be "trivial" in the diverse case, implying that the inclusion is also singular. Then we prove that the inclusion

L (F_{d}) s u b s e t e q L (m a t h s c r T_{d} (G_{*}))

satisfies the weak asymptotic homomorphism property, which yields another proof that the inclusion is singular. Finally, we finish the paper with an application: Using irreducibility, our conditions for when

L (m a t h s c r T_{d} (G_{*}))

is a type

e x t {I I}_{1}

McDuff factor, and the fact that

F_{d}

is character rigid (in the sense of Peterson), we prove that the groups

F_{d}

are McDuff (in the sense of Deprez-Vaes).

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