REMARKS ON THE DIAGONAL EMBEDDING AND STRONG 1-BOUNDEDNESS SRIVATSAV KUNNAWALKAM ELAYAVALLI

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REMARKS ON THE DIAGONAL EMBEDDING AND STRONG

1-BOUNDEDNESS

SRIVATSAV KUNNAWALKAM ELAYAVALLI

Abstract. We identify a large class of hyperbolic groups whose von Neumann algebras are not

strongly 1-bounded: Sela’s hyperbolic towers over F2subgroups. We also show that any inter-

mediate subalgebra of the diagonal embedding of L(F2) into its ultrapower doesn’t have Property

(T).

1. Introduction

Voiculescu initiated a revolutionary theory of free entropy in his paper [Voi94]. The free entropy

computes the asymptotic volume of the microstate spaces (matrix models approximating the dis-

tribution of a ﬁxed tuple in a tracial von Neumann algebra). Voiculescu’s asymptotic freeness

theorem allowed him to show that the free entropy of a tuple of freely independent semicirculars is

non vanishing. On the other hand, one is able to compute the free entropy when there are algebraic

constraints present in the ambient algebra, such as suﬃciently many commutation relations or the

existence of diﬀuse regular subalgebras that are hyperﬁnite. Combining these two ideas, Voiculescu

(in [Voi96]) showed that L(F2) admits no Cartan subalgebras, then Ge (in [Ge98]) obtained using

the same idea, that L(F2) is prime. These settled problems left open by Popa in [Pop83] where he

showed that L(FX) is prime and admits no Cartan subalgebras where Xis an uncountable set.

One of the main modern threads of Voiculescu’s free entropy theory is that of strong 1-boundedness

for von Neumann algebras, which originated with remarkable ideas of Jung in [Jun07]. Inspired

by ideas from geometric measure theory, in particular Besicovitch’s classiﬁcation of metric spaces

with Hausdorﬀ measure 1, Jung developed technical tools to study the case when Voiculescu’s free

entropy dimension (a Minkowski dimension type quantity for the microstate spaces) for a tuple

is 1, and discovered natural conditions wherein this property passes to the von Neumann algebra

generated by the tuple. In particular, if one locates such a tuple in a von Neumann algebra, one

can automatically conclude non isomorphism with L(F2), as the free entropy dimension of the

semicircular generating set is 2. Jung used this in [Jun07] to prove that L(F2) cannot be generated

by two amenable (more generally strongly 1-bounded) subalgebras with diﬀuse intersection.

More recently, by carefully analyzing ideas of Jung, Hayes in [Hay18] extracted a numerical invariant

of the von Neumann algebra, called the 1-bounded entropy. This framework has proved very robust

and has been used to obtain several new rigidity results for non strongly 1-bounded von Neumann

algebras (such as L(F2)). For instance, Hayes showed in [Hay18] that L(F2) does not admit even

quasi regular diﬀuse strongly 1-bounded subalgebras, generalizing the Theorem of Voiculescu. For

more recent results see [HJNS21, Hay20, HJKE22, BC22, CIE22]. For these reasons it is of great

interest to identify examples of non strongly 1-bounded von Neumann algebras.

In this note we observe using 1-bounded entropy, some structural properties of intermediate subal-

gebras of the diagonal embedding of L(F2) into its ultrapower. By way of leveraging existentially

closed (see Section 2) copies of F2in the group level, our ﬁrst main result identiﬁes a family of

S.K.E was supported by a Simons Postdoctoral Fellowship.

arXiv:2210.03783v2 [math.OA] 23 Mar 2023

2 SRIVATSAV KUNNAWALKAM ELAYAVALLI

hyperbolic groups introduced by Sela ( [Sel01]) whose von Neumann algebras are not strongly

1-bounded von Neumann algebras. Some familiar examples are hyperbolic surface groups.

Theorem A. The group von Neumann algebras of all hyperbolic towers over F2subgroups are not

strongly 1-bounded.

Remark 1.1. We thank D. Shlyakhtenko for pointing out to us that surface group von Neumann

algebras are not strongly 1-bounded is already known through a computation of the free entropy di-

mension, which is an apriori stronger result (see paragraph below [BDJ08] Theorem 4.13). Roughly

speaking, one sees that the hyperbolic surface groups are decomposed as an iterated amalgamated

free product over copies of Z. Then, using the free entropy dimension estimate for amalgamated

free products over hyperﬁnite subalgebras, which is the main technical result of [BDJ08] (Theorem

4.4), one can identify a generating set using an iterative process, whose microstates free entropy

dimension has a precise lower bound (in the case of genus g, the lower bound is 2g−1 which is

signiﬁcantly greater than 1). We would like to point out that our proof not only applies to more

groups but is conceptually diﬀerent and softer. Indeed, on the von Neumann algebra level we only

use the fact that L(F2) is not strongly 1-bounded, and the 1-bounded entropy inequality (see Fact

5.2) whose proof as outlined in [Hay18] Proposition 4.5 is quite elementary.

Potentially, there is a larger class of groups that admit an existential copy of F2, however in

light of Sela’s classiﬁcation of groups that are elementarily equivalent to free groups, ﬁnding more

examples could be hard. Note also that there also is the famous class of existentially closed groups

(see [HS88]). We document a proof that the group von Neumann algebras of these groups are on

the other hand McDuﬀ II1factors (see Proposition 3).

By virtue of being hyperbolic, many rigidity results are already known in the setting of Theorem A

through Ozawa’s biexactness techniques and Popa’s deformation rigidity (see [Oza04,OP10,PV14,

CS13]). However, one obtains using non-strong 1-boundedness, the following stronger rigidity

results below:

Corollary B. Let Nbe the group von Neumann algebra of a hyperbolic tower over an F2subgroup.

Then the following hold:

(1) Ncannot be written as the join of two strongly 1-bounded subalgebras (see a comprehensive

list in 5.5) with diﬀuse intersection.

(2) Ncontains no diﬀuse quasi regular strongly 1-bounded subalgebra.

Our next observation describes a surprising structural property of intermediate subalgebras of the

diagonal embedding:

Theorem C. For any ultraﬁlter Uon any set I, one cannot embed a property (T) von Neumann

algebra Minto L(F2)Usuch the Mcontains the diagonal copy of L(F2).

We would like to point out that Theorem C works if we just replace L(F2) by any ﬁnite von

Neumann algebra Nwith h(N∩M:N) = ∞(see the ﬁrst paragraph of Section 5). Also note

that any Mhere (regardless of whether Mhas property (T) or not) will also not have any diﬀuse

quasi regular amenable subalgebra. We record below a question asked to us by J. Peterson, as a

conjecture:

Conjecture 1.2. Let Nbe a non Gamma II1factor with the Haagerup property. Then for any

ultraﬁlter Uon any set I, one cannot embed a property (T) von Neumann algebra Minto NU

containing the diagonal embedding.

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REMARKSONTHEDIAGONALEMBEDDINGANDSTRONG1-BOUNDEDNESSSRIVATSAVKUNNAWALKAMELAYAVALLIAbstract.WeidentifyalargeclassofhyperbolicgroupswhosevonNeumannalgebrasarenotstrongly1-bounded:Sela'shyperbolictowersoverF2subgroups.Wealsoshowthatanyinter-mediatesubalgebraofthediagonalembeddingofL(F2)intoitsultrapower...

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REMARKS ON THE DIAGONAL EMBEDDING AND STRONG 1-BOUNDEDNESS SRIVATSAV KUNNAWALKAM ELAYAVALLI

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