# The linear $\operatorname{SL}_{2}(\mathbb{Z})$-action on $\mathbb{T}^{n}$:
ergodic and von Neumann algebraic aspects

Paul JOLISSAINT and Alain VALETTE

###### Abstract

The unique irreducible representation of $\operatorname{SL}_{2}(\mathbb{R})$
on $\mathbb{R}^{n}$ induces an action, called the linear action, of
$\operatorname{SL}_{2}(\mathbb{Z})$ on the torus $\mathbb{T}^{n}$ for every
$n\geq 2$. For $n$ odd, it factors through
$\operatorname{PSL}_{2}(\mathbb{Z})$, so we denote by $G_{n}$ the group
$\operatorname{SL}_{2}(\mathbb{Z})$ for $n$ even, and
$\operatorname{PSL}_{2}(\mathbb{Z})$ for $n$ odd. We prove that the action is
free and ergodic for every $n\geq 2$, that if
$h\in\operatorname{SL}_{2}(\mathbb{Z})$ is a hyperbolic element and if $n$ is
even, then the action of the subgroup generated by $h$ is still ergodic, but
also that, for $n$ odd, no amenable subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$ acts ergodically on $\mathbb{T}^{n}$. We
deduce also that every ergodic sub-equivalence relation $\mathcal{R}$ of the
orbital equivalence relation $\mathcal{S}_{n}$ of $G_{n}$ on $\mathbb{T}^{n}$
is either amenable or rigid, extending a result by Ioana for $n=2$. This
result has the following corollaries: firstly, for $n\geq 2$ even, if $H$ is a
maximal amenable subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$ containing an
hyperbolic matrix, then the associated crossed product II1 factor
$L^{\infty}(\mathbb{T}^{n})\rtimes H$ is a maximal Haagerup subalgebra of
$L^{\infty}(\mathbb{T}^{n})\rtimes\operatorname{SL}_{2}(\mathbb{Z})$; secondly
, for every $n$, the fundamental group of $L^{\infty}(\mathbb{T}^{n})\rtimes
G_{n}$ is trivial.

##  1 Introduction

It is a classical fact that, for $n\geq 1$, the Lie group
$\operatorname{SL}_{2}(\mathbb{R})$ admits a unique irreducible representation
on $\mathbb{R}^{n}$. Indeed, let $P_{n-1}(\mathbb{R})$ be the space of
homogeneous polynomials of degree $n-1$ in two variables $X,Y$: we have a
standard representation $\rho_{n-1}$ of $\operatorname{SL}_{2}(\mathbb{R})$ on
$P_{n-1}(\mathbb{R})$ defined by

$\Big{(}\rho_{n-1}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{)}(P)(X,Y)=P(aX+cY,bX+dY)$

for $P\in P_{n-1}(\mathbb{R})$ and $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{R})$. Now we may identify
$P_{n-1}(\mathbb{R})$ with $\mathbb{R}^{n}$ by means of the canonical basis
$X^{n-1},X^{n-2}Y,\ldots,XY^{n-2},Y^{n-1}$. It is clear that the subgroup
$P_{n-1}(\mathbb{Z})$ of polynomials with integer coefficients is invariant
under $\rho_{n-1}(\operatorname{SL}_{2}(\mathbb{Z}))$, so we get an
$\operatorname{SL}_{2}(\mathbb{Z})$-action on the $n$-dimensional torus
$\mathbb{T}^{n}=P_{n-1}(\mathbb{R})/P_{n-1}(\mathbb{Z})$. If $n$ is odd, the
representation $\rho_{n-1}$ factors through
$\operatorname{PSL}_{2}(\mathbb{R})$, so the action of
$\operatorname{SL}_{2}(\mathbb{Z})$ on $\mathbb{T}^{n}$ factors through
$\operatorname{PSL}_{2}(\mathbb{Z})$.

Define $G_{n}$ as $\operatorname{SL}_{2}(\mathbb{Z})$ for $n$ even and as
$\operatorname{PSL}_{2}(\mathbb{Z})$ for $n$ odd. In this paper, by linear
action on $\mathbb{T}^{n}$ we mean the action of $G_{n}$ on $\mathbb{T}^{n}$.
Observe that the linear action preserves the normalized Lebesgue measure on
$\mathbb{T}^{n}$, so it is a p.m.p. (probability measure-preserving) action.
For $n=2$, we recover the well-studied action of
$\operatorname{SL}_{2}(\mathbb{Z})$ on the 2-torus. Our first result
generalizes a well-known property of this case:

###### Theorem 1.1

Fix $n\geq 2$.

  1. 1.

The linear action on $\mathbb{T}^{n}$ is ergodic and essentially free.

  2. 2.

If $h\in\operatorname{SL}_{2}(\mathbb{Z})$ is hyperbolic and $n$ is even, then
the subgroup $\langle h\rangle$ generated by $h$ acts ergodically on
$\mathbb{T}^{n}$.

We observe that the second part of Theorem 1.1 only holds for $n$ even. Indeed
we show:

###### Proposition 1.2

For $n$ odd, no amenable subgroup of $\operatorname{PSL}_{2}(\mathbb{Z})$ acts
ergodically on $\mathbb{T}^{n}$.

A countable, ergodic, measure preserving equivalence relation $\mathcal{R}$ on
a standard probability space $(X,\mu)$ is said to be rigid if the inclusion of
$L^{\infty}(X,\mu)$ into the von Neumann algebra $L(\mathcal{R})$ associated
with $\mathcal{R}$ (see [FM77II]), is rigid in the sense of Popa [Po06]. In
Theorem 0.1 of [Io10], Ioana discovered a remarkable property of the orbital
equivalence relation $\mathcal{S}_{2}$ induced by the linear action of
$\operatorname{SL}_{2}(\mathbb{Z})$ on $\mathbb{T}^{2}$, namely: every ergodic
sub-equivalence relation of $\mathcal{S}_{2}$ is either amenable of rigid
111This is a vast generalization of a result by Burger [Bur91] that says that,
for $H$ a subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$, either $H$ is
amenable (and so $\mathbb{Z}^{2}\rtimes H$ is amenable), or the pair
$(\mathbb{Z}^{2}\rtimes H,\mathbb{Z}^{2})$ has the relative property (T).. We
will refer to this phenomenon as Ioana’s dichotomy. More examples of this
dichotomy, for homogeneous spaces of certain Lie groups, were provided by
Ioana and Shalom (Theorem C in [IS13]). We contribute to the list by showing:

###### Theorem 1.3

For $n\geq 2$, let $\mathcal{S}_{n}$ be the orbital equivalence relation on
$\mathbb{T}^{n}$ induced by the linear action. Then Ioana’s dichotomy holds
for $\mathcal{S}_{n}$.

In the final part of the paper, we consider von Neumann algebraic aspects,
more precisely we deal with the von Neumann algebra $N_{n}\coloneqq
L^{\infty}(\mathbb{T}^{n})\rtimes G_{n}$ obtained via the group measure space
construction. It is a type II1-factor as a consequence of Theorem 1.1. Denote
by $L(G)$ the group von Neumann algebra of a countable group $G$; using
Fourier series we may identify $N_{n}$ with $L(\mathbb{Z}^{n}\rtimes G_{n})$.
Now $G_{n}$ has the Haagerup property and the pair $(\mathbb{Z}^{n}\rtimes
G_{n},\mathbb{Z}^{n})$ has the relative property (T) (by Proposition 3.1 in
[Va23]), so the factors $N_{n}$ belong to Popa’s $\mathcal{HT}$-class [Po06];
hence by section 6 and Corollary 8.2 of [Po06] they inherit $\ell^{2}$-Betti
numbers $\beta_{k}^{HT}(N_{n})$ that coincide with those of the acting group
$G_{n}$, and we have:

###### Theorem 1.4

For every $n\geq 2$, the $\ell^{2}$-Betti numbers of the factor $N_{n}$
satisfy $\beta_{k}^{HT}(N_{n})=0$ for every $k\not=1$ and

$\beta_{1}^{HT}(N_{n})=\begin{cases}1/12&\textrm{for\ even\ }n\\\
1/6&\textrm{for\ odd\ }n.\end{cases}$

In particular, if $n$ is odd and $m$ is even, then $N_{n}$ and $N_{m}$ are
non-isomorphic, and, for every $n$, the fundamental group $\mathcal{F}(N_{n})$
is trivial.

In Theorem 3.1 of [JS21], Jiang and Skalski proved that, if $H$ is a maximal
amenable subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$ containing some
hyperbolic matrix 222By Proposition 2.6 in [Va23], the subgroup $H$ is
isomorphic either to $\mathbb{Z}\times C_{2}$ or to $\mathbb{Z}\rtimes
C_{4}$., then $L^{\infty}(\mathbb{T}^{2})\rtimes H$ is a maximal Haagerup von
Neumann subalgebra of $N_{2}$, where the Haagerup property for finite von
Neumann algebras has been studied first in [Cho83] and then systematically by
the first author in [Jo02]. Building on Theorem 1.3, we extend that result as
follows:

###### Theorem 1.5

Fix $n\geq 2$. Then for any hyperfinite subfactor $P$ of $N_{n}$ which
contains $L^{\infty}(\mathbb{T}^{n})\cong L(\mathbb{Z}^{n})$ as a Cartan
subalgebra, every maximal Haagerup von Neumann subalgebra $P\subset Q\subset
N_{n}$ is a hyperfinite subfactor of $N_{n}$. Moreover:

  1. 1.

If $n$ is even, $Q$ can be chosen of the form $Q=L(\mathbb{Z}^{n}\rtimes H)$
where $H$ is a maximal amenable subgroup of
$\operatorname{SL}_{2}(\mathbb{Z})$ which contains a hyperbolic element.

  2. 2.

If $n$ is odd, $P$ and $Q$ are not of the form $L(K)$ for any subgroup $K$ of
$\mathbb{Z}^{n}\rtimes\operatorname{PSL}_{2}(\mathbb{Z})$.

The existence of a hyperfinite subfactor $P$ as in the preceeding theorem is a
consequence of Proposition 3.6 of [Po81]. Moreover, in a finite von Neumann
algebra, every von Neumann subalgebra with the Haagerup property is contained
in a maximal one: we provide a proof of this fact in Proposition 4.6
below.333This fact is stated in Corollary 1.12 of [JS21] as a consequence of
Lemma 1.11 of the same article. However, the latter only treats increasing
unions of countably many subalgebras, which is not sufficient to apply Zorn’s
lemma.

By a celebrated result of Connes-Feldman-Weiss [CFW81], the sub-equivalence
relation $\mathcal{R}$ of $\mathcal{S}_{n}$ such that $P=L(\mathcal{R})$ is
induced by a $\mathbb{Z}$-action. Part 2 of Theorem 1.5 above shows that
$\mathcal{R}$ will not be produced by any subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$.

Section 2 contains the proofs of Theorem 1.1 and Proposition 1.2. Section 3 is
about Ioana’s dichotomy: the main new ingredient is the fact that the
$\operatorname{SL}_{2}(\mathbb{R})$-action induced on the projective space
$\mathbb{P}^{n-1}(\mathbb{R})$ by the representation $\rho_{n-1}$, is amenable
(see Proposition 3.6): for $n=2$, i.e. the case treated in [Io10], it is clear
that the group $\operatorname{SL}_{2}(\mathbb{R})$ acts amenably on
$\mathbb{P}^{1}(\mathbb{R})$, as $\mathbb{P}^{1}(\mathbb{R})$ identifies with
$\operatorname{SL}_{2}(\mathbb{R})/B$, where $B$ is the Borel subgroup, and
$B$ is amenable (see Proposition 4.3.2 in [Zi84]). The final section 4
contains the proofs of Theorems 1.4 and 1.5.

Acknowledgements: We thank D. Gaboriau, A. Ioana, N. Monod and S. Vaes for
useful exchanges.

##  2 Ergodicity

Proof of Theorem 1.1:

  1. 1.

We start with essential freeness. Observe that
$\rho_{n-1}(\operatorname{SL}_{2}(\mathbb{R}))\subset\operatorname{SL}_{n}(\mathbb{R})$
(otherwise, composing $\rho_{n-1}$ with the determinant we would get a non-
trivial homomorphism from $\operatorname{SL}_{2}(\mathbb{R})$ to an abelian
group); hence
$\rho_{n-1}(\operatorname{SL}_{2}(\mathbb{Z}))\subset\operatorname{SL}_{n}(\mathbb{R})\cap
N_{n}(\mathbb{Z})=\operatorname{SL}_{n}(\mathbb{Z})$, so that the linear
action on $\mathbb{T}^{n}$ is the restriction to
$\rho_{n-1}(\operatorname{SL}_{2}(\mathbb{Z}))$ of the standard action of
$\operatorname{SL}_{n}(\mathbb{Z})$ on $\mathbb{T}^{n}$. The latter is known
to be essentially free, see e.g. lemma 5.2.4 in [Zi84] and the comment
following it.

Concerning ergodicity, view $\mathbb{T}^{n}$ as a compact abelian group on
which $\operatorname{SL}_{2}(\mathbb{Z})$ acts by automorphisms. View
$\mathbb{Z}^{n}$ as the Pontryagin dual of $\mathbb{T}^{n}$. By Proposition
1.5 in [BM00], the linear action is ergodic if and only if every non-zero
$\operatorname{SL}_{2}(\mathbb{Z})$-orbit on $\mathbb{Z}^{n}$ is infinite. So
assume by contradiction that some non-zero $v\in\mathbb{Z}^{n}$ has a finite
orbit. Then the stabilizer $\Gamma$ of $v$ has finite index in
$\operatorname{SL}_{2}(\mathbb{Z})$, in particular $\Gamma$ is non-amenable.
By Proposition 3.1 in [Va23], the restriction of $\rho_{n-1}$ to $\Gamma$ is
irreducible, so it cannot have non-zero invariant vectors, and we reached a
contradiction.

  2. 2.

By Example 1.4.(iii) in [BM00], it is enough to check that none of the complex
eigenvalues of $\rho_{n-1}(h)$ is a root of unit. But for some
$\lambda\in\mathbb{R}$ with $|\lambda|>1$, the matrix $h$ is conjugate in
$\operatorname{SL}_{2}(\mathbb{R})$ to
$h_{\lambda}=\begin{pmatrix}\lambda&0\\\ 0&1/\lambda\end{pmatrix}$. Now
$\rho_{n-1}(h_{\lambda})$ is diagonal in the basis
$X^{n-1},X^{n-2}Y,\ldots,XY^{n-2},Y^{n-1}$ of $P_{n-1}(\mathbb{R})$, with
eigenvalues
$\lambda^{n-1},\lambda^{n-3},\ldots,\lambda^{-(n-3)},\lambda^{-(n-1)}$. Since
$n$ is even, none of those eigenvalues is a root of 1. $\blacksquare$

Proof of Proposition 1.2: Fix some odd $n$. Let $H$ be an amenable subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$, we must show that $H$ has a finite orbit
in $\mathbb{Z}^{n}\setminus\\{0\\}$. Clearly we may assume that $H$ is maximal
amenable, so we may appeal to the classification in Lemma 2.5 of [Va23],
giving three cases to consider:

  * •

$H$ is cyclic of order $3$, and there is nothing to prove.

  * •

$H$ is infinite cyclic consisting of parabolic elements: then $H$ is conjugate
to the subgroup generated by $u=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$; but
$\rho_{n-1}(u)$ fixes the monomial $X^{n-1}$.

  * •

$H$ is either infinite cyclic or infinite dihedral, with torsion-free part
being generated by some hyperbolic matrix $h$, conjugate in
$\operatorname{PSL}_{2}(\mathbb{R})$ to a diagonal matrix
$h_{\lambda}=\begin{pmatrix}\lambda&0\\\ 0&1/\lambda\end{pmatrix}$ with
$\lambda>1$. As in the proof of the second part of Theorem 1.1, the
eigenvalues of $\rho_{n-1}(h_{\lambda})$ (hence also of $\rho_{n-1}(h)$) are
$\lambda^{n-1},\lambda^{n-3},\ldots,\lambda^{-(n-3)},\lambda^{-(n-1)}$, but
now $1$ appears in the list, as $n$ is odd. So $\rho_{n-1}(h)$ admits in
$\mathbb{R}^{n}$ an eigenvector $v$ with eigenvalue $1$. As $\rho_{n-1}(h)$
has integer coefficients, by Gaussian elimination we may assume $v$ to have
rational coordinates, and by chasing denominators we may assume
$v\in\mathbb{Z}^{n}$. Then the orbit of $v$ under $\rho_{n-1}(H)$ has at most
2 elements. $\blacksquare$

##  3 Ioana’s dichotomy

###  3.1 Equivalence relations on standard spaces

Let $(X,\mathcal{B},\mu)$ be a standard probability space. A (countable)
standard equivalence relation on $X$ is a Borel subset $\mathcal{R}\subset
X\times X$ which is an equivalence relation whose equivalence classes
$[x]_{\mathcal{R}}\coloneqq\\{y\in X\colon x\sim y\\}$ are all at most
countable. We denote by $p^{i}:X\times X\rightarrow X$ the projection defined
by $p^{i}(x_{1},x_{2})=x_{i}$ for $i=1,2$; as they are countable-to-one maps,
they imply the existence of $\sigma$-finite measures $\nu_{i}$, $i=1,2$, on
the Borel subsets of $\mathcal{R}$ which are defined by

$\nu_{i}(C)\coloneqq\int_{X}|(p^{i})^{-1}(x)\cap C|d\mu(x)$

for every such Borel set $C$. We assume henceforth that $\mu$ is invariant
under $\mathcal{R}$, which means that $\nu\coloneqq\nu_{1}=\nu_{2}$. Thus, for
every integrable function $f$ on $\mathcal{R}$, one has

$\int_{\mathcal{R}}f(x,y)d\nu(x,y)=\int_{X}\Big{(}\sum_{x\sim
y}f(x,y)\Big{)}d\mu(x)=\int_{X}\Big{(}\sum_{y\sim x}f(x,y)\Big{)}d\mu(y).$

Finally, the full group of $\mathcal{R}$, denoted by $[\mathcal{R}]$, consists
in all automorphisms $\theta$ of $(X,\mathcal{B},\mu)$ such that
$\theta(x)\in[x]_{\mathcal{R}}$ for almost every $x\in X$, and $\mathcal{R}$
is ergodic if any $\mathcal{R}$-invariant Borel subset of $X$ is either null
or co-null.

Natural examples of countable equivalence relations are given by p.m.p.
actions of countable groups: if $\Gamma\curvearrowright(X,\mu)$ is such an
action, then the orbital equivalence relation
$\mathcal{R}_{\Gamma}\coloneqq\\{(x,\gamma x)\colon x\in X,\gamma\in\Gamma\\}$
is a countable equivalence relation which is ergodic if and only if the action
of $\Gamma$ is. It turns out that, by Theorem 3 of [FM77I], every equivalence
relation is of this form, but the acting group is far from being unique.

Recall that $\mathcal{R}$ is hyperfinite if there exists a Borel automorphism
$T$ of $(X,\mu)$ such that, for (almost) every $x\in X$, the equivalence class
$[x]_{\mathcal{R}}$ coincides with the orbit $\\{T^{k}x\colon
k\in\mathbb{Z}\\}$ of $x$. It turns out that, by the main result of [CFW81],
hyperfiniteness of $\mathcal{R}$ is equivalent to its amenability, whose
definition we recall from [CFW81]:

###### Definition 3.1

With $X,\mathcal{R}$ as above: say that $\mathcal{R}$ is an amenable
equivalence relation if there is an assignment $x\mapsto\nu_{x}$ from $X$ to
the state space of $\ell^{\infty}([x]_{\mathcal{R}})$ such that:

  1. (1)

the family $(\nu_{x})_{x\in X}$ is $\mathcal{R}$-invariant, i.e.
$\gamma_{*}(\nu_{x})=\nu_{y}$ for every $\gamma=(y,x)\in\mathcal{R}$;

  2. (2)

for every $f\in L^{\infty}(\mathcal{R},\nu)$, the function
$x\mapsto\nu_{x}(f)$ is measurable.

###  3.2 Amenability of the linear action of
$\operatorname{SL}_{2}(\mathbb{R})$ on $\mathbb{P}^{n-1}(\mathbb{R})$

Let $G$ be a locally compact, second countable group. First we recall the
definition of an amenable group action of $G$ on a standard measure space,
borrowed from Definition 4.3.1 in [Zi84].

###### Definition 3.2

Let $G$ act measurably on a standard measure space $(X,\mu)$ by preserving the
class of $\mu$. The action of $G$ on $X$ is amenable if every affine space
over $X$ with an $\alpha$-twisted action, where $\alpha$ denotes the
corresponding cocycle, admits a $G$-fixed point.

Important for us will be the following result by Adams, Elliott and Giordano
(Theorem 5.1 in [AEG94]):

###### Theorem 3.3

Assume that $G$ acts measurably on a standard measure space $(X,\mu)$ by
preserving the class of $\mu$. If stabilizers are amenable $\mu$-almost
everywhere, and if the orbital equivalence relation $\mathcal{R}_{G}$ is
amenable, then the action of $G$ on $X$ is amenable. $\blacksquare$

Recall (see Definition 2.1.9 in [Zi84]) that the action of $G$ on $X$ is
smooth if the quotient Borel structure on $G\backslash X$ is countably
separated. The next result is probably well-known, but we were unable to find
a proof in the literature.

###### Lemma 3.4

Let $G$ act on $(X,\mu)$ as above. If $G$ acts smoothly on $X$, then the
orbital equivalence relation $\mathcal{R}_{G}$ induced by $G$ on $X$, is
amenable.

Proof: We proceed in 4 steps.

  * •

By a result of Feldman, Hahn and Moore (Theorem 2.8 in [FHM78]), there exists
a Borel subset $Y\subset X$, with $G\cdot Y$ being co-null, such that $G\cdot
x\cap Y$ is countable for every $x\in X$. We denote by $\mathcal{R}_{Y}$ the
restriction to $Y$ of the relation $\mathcal{R}_{G}$, so that
$\mathcal{R}_{Y}$ is an equivalence relation with countable classes.

  * •

By Corollary 16 in [CFW81], the amenability of $\mathcal{R}_{G}$ is equivalent
to that of $\mathcal{R}_{Y}$.

  * •

As $G\backslash X$ identifies with $\mathcal{R}_{Y}\backslash Y$ (up to a null
set), the Borel structure on $\mathcal{R}_{Y}\backslash Y$ is countably
separated. Let

$p:Y\rightarrow\mathcal{R}_{Y}\backslash Y:y\mapsto[y]_{\mathcal{R}_{Y}}$

be the quotient map. By von Neumann’s selection theorem (see [Zi84], Theorem
A.9), there exists a measurable map $s:\mathcal{R}_{Y}\backslash Y\rightarrow
Y$ such that $p(s[y]_{\mathcal{R}_{Y}})=[y]_{\mathcal{R}_{Y}}$ for almost
every $[y]_{\mathcal{R}_{Y}}\in\mathcal{R}_{Y}\backslash Y$.

  * •

Finally, referring to Definition 3.1, we construct the assignment
$y\mapsto\nu_{y}$ from $Y$ to the state space of
$\ell^{\infty}([y]_{\mathcal{R}})$ by defining $\nu_{y}$ as the Dirac mass
$\delta_{(y,s[y])}$ at $(y,s[y])\in\mathcal{R}_{Y}$; the first condition is
then clear, and the second amounts to the measurability of $y\mapsto
f(y,s[y])$, which is clear as well. $\blacksquare$

We now deal with the action of $\operatorname{SL}_{2}(\mathbb{R})$ on the
projective space $\mathbb{P}^{n-1}(\mathbb{R})$ associated with the
representation $\rho_{n-1}$; it preserves the class of Lebesgue measure on
$\mathbb{P}^{n-1}(\mathbb{R})$.

###### Lemma 3.5

For $n\geq 2$, the stabilizers of points of $\mathbb{P}^{n-1}(\mathbb{R})$ in
$\operatorname{SL}_{2}(\mathbb{R})$, are amenable.

Proof: Since $\rho_{n-1}$ is an irreducible representation, it leaves no
1-dimensional linear subspace of $\mathbb{R}^{n}$ invariant. So the stabilizer
of a point in $\mathbb{P}^{n-1}(\mathbb{R})$ is a proper algebraic subgroup of
$\operatorname{SL}_{2}(\mathbb{R})$, hence of dimension $<3$, hence amenable.
$\blacksquare$

###### Proposition 3.6

For $n\geq 2$, the action of $\operatorname{SL}_{2}(\mathbb{R})$ induced on
the projective space $\mathbb{P}^{n-1}(\mathbb{R})$ by the representation
$\rho_{n-1}$, is amenable.

Proof: We check the assumptions of Theorem 3.3 above. Amenability of
stabilizers is Lemma 3.5. On the other hand,
$\operatorname{SL}_{2}(\mathbb{R})$ is a real algebraic group acting
algebraically on the variety $\mathbb{P}^{n-1}(\mathbb{R})$, so orbits are
locally closed (see Theorem 3.1.3 in [Zi84]). This implies that the
$\operatorname{SL}_{2}(\mathbb{R})$-action on $\mathbb{P}^{n-1}(\mathbb{R})$
is smooth (Proposition 2.1.12 in [Zi84]); by lemma 3.4, the relation
$\mathcal{R}_{G}$ is amenable. So Theorem 3.3 applies to give the conclusion.
$\blacksquare$

It was already mentioned in the introduction that Proposition 3.6 is easy to
prove for $n=2$. We observe that there is also a simple proof for $n=3$.

###### Example 3.7

It is well-known that $\rho_{2}(\operatorname{SL}_{2}(\mathbb{R}))$ can be
identified with $SO_{o}(2,1)$, the connected component of identity of the
special orthogonal group of the quadratic form $x^{2}+y^{2}-z^{2}$ on
$\mathbb{R}^{3}$. Now $G=SO_{o}(2,1)$ has 3 orbits in the projective plane
$\mathbb{P}^{2}(\mathbb{R})$: the conic $C\equiv x^{2}+y^{2}-z^{2}=0$, that
identifies with $G/B$ (with $B$ the Borel subgroup); the interior of $C$, that
identifies with $G/K$ (with $K$ a maximal compact subgroup); and the exterior
of $C$, that identifies with $G/A$ (with $A$ a maximal split torus). As
$B,K,A$ are all amenable, the $G$-action on $\mathbb{P}^{2}(\mathbb{R})$ is
amenable.

###  3.3 Rigidity for inclusions and for equivalence relations

In the present article, all von Neumann algebras are assumed to have separable
preduals, and all considered tracial states are assumed to be normal and
faithful; they will be also called traces for short. Let then $N$ be a finite
von Neumann algebra equipped with a trace $\tau$, and $1\in B\subset N$ be a
von Neumann subalgebra of $N$ with the same unit. Recall that $N$ identifies
with a dense subspace $\hat{N}$ of the Hilbert space $L^{2}(N,\tau)$ so that
$\langle\hat{x},\hat{y}\rangle=\tau(y^{*}x)$ for all $x,y\in N$; recall also
that the map $J:\hat{x}\mapsto\widehat{x}^{*}$ extends to an antilinear
isometry on $L^{2}(N,\tau)$ such that the operator $Jy^{*}J\in
B(L^{2}(N,\tau))$ acts on the right: $Jy^{*}J\hat{x}=\widehat{xy}$ for all
$x,y\in N$.

We will consider below the following particular class of von Neumann
subalgebras of $N$: a Cartan subalgebra $A$ of $N$ is a maximal abelian von
Neumann subalgebra of $N$ whose normalizer $\mathcal{N}_{N}(A)\coloneqq\\{u\in
U(N)\colon uAu^{*}=A\\}$ generates $N$.

Important von Neumann algebras that will be considered here are the associated
von Neumann algebras of groups: let $G$ be a countable group; its von Neumann
algebra $L(G)$ is generated by the left regular representation $g\mapsto
u_{g}$ on $\ell^{2}(G)$. It is finite, and it is equipped with the natural
trace $\tau$ defined by $\tau(x)=\langle x\delta_{e},\delta_{e}\rangle$ for
all $x\in L(G)$. Every $x\in L(G)$ has a unique ”Fourier series” expansion
$\sum_{g}x(g)u_{g}$, where $x(g)=\tau(xu_{g^{-1}})$ and
$\sum_{g}|x(g)|^{2}=\|x\|_{2}^{2}$. If $H<G$ is a subgroup, then $L(H)$
identifies in a natural way with a von Neumann subalgebra of $L(G)$, its
elements $y$ being characterized by the fact that $y(g)=0$ for all $g\in
G\setminus H$. Recall also that $L(G)$ is a factor if and only if $G$ is icc,
i.e. it has infinite conjugacy classes, except the class of $e$.

More generally, assume that there is a trace-preserving action $\sigma$ of a
group $G$ on a finite von Neumann algebra $(B,\tau)$. Then the associated
crossed product is a finite von Neumann algebra acting on
$L^{2}(B)\otimes\ell^{2}(G)$; it is generated by a copy of $B$ and a unitary
representation $g\mapsto u_{g}$ of $G$ in such a way that $u$ is quasi-
equivalent to the regular representation and that
$u_{g}bu_{g}^{*}=\sigma_{g}(b)$ for all $g\in G$ and $b\in B$. It is denoted
by $B\rtimes_{\sigma}G$.

Let us now recall the definition of von Neumann algebras associated to
equivalence relations as in [FM77II].

Let $\mathcal{R}$ be a standard equivalence relation and let $s$ be a
normalized $2$-cocycle on $\mathcal{R}$: the latter is a Borel
$\mathbb{T}$-valued map defined on

$\mathcal{R}^{2}\coloneqq\\{(x,y,z)\in X^{3}\colon x\sim y\sim z\\}$

which has the following properties:

  1. (a)

$s(x,y,z)=1$ as soon as $|\\{x,y,z\\}|\leq 2$;

  2. (b)

for all $x_{1}\sim x_{2}\sim x_{3}\sim x_{4}$,

$s(x_{1},x_{3},x_{4})s(x_{1},x_{2},x_{3})=s(x_{1},x_{2},x_{4})s(x_{2},x_{3},x_{4}).$

The von Neumann algebra $L(\mathcal{R},s)$ acts on
$\mathcal{H}=L^{2}(\mathcal{R},\nu)$ and it is generated by the following two
families of operators:

  1. (1)

We associate to $\phi\in[\mathcal{R}]$ the operator $u_{\phi}$ on
$\mathcal{H}$ defined by

$(u_{\phi}\xi)(x,z)=\xi(\phi^{-1}(x),z)s(x,\phi^{-1}(x),z)=\sum_{y\sim
x}\delta_{y,\phi^{-1}(x)}\xi(y,z)s(x,y,z);$

  2. (2)

next, we associate to every $a\in L^{\infty}(X,\mu)$ the multiplication
operator on $\mathcal{H}$, still denoted by $a$, which is defined by

$(a\xi)(x,z)=a(x)\xi(x,z)(=a(x)\xi(x,z)\underbrace{s(x,x,z)}_{=1}).$

These two families of operators satisfy the following relations which are
straightforward consequences of the definitions and whose proofs are left to
the reader:

###### Lemma 3.8

Retain the above definitions. Then

  1. (A)

If $\theta,\phi\in[\mathcal{R}]$, then
$u_{\theta}u_{\phi}=\sigma_{\theta,\phi}\cdot u_{\theta\phi}$ where
$\sigma_{\theta,\phi}\in L^{\infty}(X,\mu)$ is the function

$\sigma_{\theta,\phi}(x)=s(x,\theta^{-1}(x),(\theta\phi)^{-1}(x)).$

In particular $u_{\theta^{-1}}u_{\theta}=1$ since
$\sigma_{\theta^{-1},\theta}(x)=s(x,\theta(x),x)=1$ for every $x$.

  2. (B)

For every $\theta\in[\mathcal{R}]$, one has $u_{\theta}^{*}=u_{\theta^{-1}}$.
Thus $u_{\theta}$ is a unitary operator for every $\theta$.

  3. (C)

For all $a\in L^{\infty}(X,\mu)$ and $\phi\in[\mathcal{R}]$, one has
$u_{\phi}au_{\phi}=a\circ\phi^{-1}$. $\blacksquare$

Thus, the set of linear combinations of the form $au_{\phi}$ is a
$*$-subalgebra of $B(\mathcal{H})$, and by definition, $L(\mathcal{R},s)$ is
its weak-operator closure, or its bicommutant, which is the same. It is a
finite von Neumann algebra equipped with the natural trace

$\tau(y)=\langle y1_{\Delta},1_{\Delta}\rangle\quad(y\in L(\mathcal{R},s))$

where $\Delta=\\{(x,x)\colon x\in X\\}$ is the diagonal of $\mathcal{R}$.

When $s=1$ is the trivial cocycle, we write $L(\mathcal{R})$ instead of
$L(\mathcal{R},1)$.

One checks that $A=L^{\infty}(X,\mu)\subset N=L(\mathcal{R},s)$ is a Cartan
subalgebra, and that $\mathcal{R}$ is ergodic if and only if
$L(\mathcal{R},s)$ is a II1 factor. Conversely, Theorem 1 of [FM77II] states
that if $A\subset N$ is a Cartan subalgebra of a finite von Neumann algebra
with separable predual, then there exists an essentially unique pair
$(\mathcal{R},s)$ on a standard probability space $(X,\mu)$ such that $A\cong
L^{\infty}(X)$ and $N=L(\mathcal{R},s)$. (In fact, $N$ need not be finite, but
we are only interested in this case here.)

Consider a p.m.p. ergodic and essentially free action
$\Gamma\curvearrowright(X,\mu)$. Then, as explained in subsection 3.1,
$\mathcal{R}_{\Gamma}$ is an ergodic equivalence relation and the inclusion
$L^{\infty}(X)\subset L(\mathcal{R}_{\Gamma})$ is naturally isomorphic to the
inclusion $L^{\infty}(X)\subset L^{\infty}(X)\rtimes\Gamma$. See for instance
section 1.2 in [Io10].

###### Remarks 3.9

(1) Observe that if $B$ is abelian and if the action is essentially free and
ergodic, then it is a Cartan subalgebra of $B\rtimes_{\sigma}G$.
(2) If $\Gamma$ is a group acting on $H$ by automorphisms, then the action
extends in a natural way to $L(H)$ and the algebra $L(H\rtimes\Gamma)$ is
naturally isomorphic to the crossed product $L(H)\rtimes\Gamma$.

We are now ready to define the relative rigidity property of pairs $B\subset
N$. In order to do that, let us recall first that a Hilbert space
$\mathcal{H}$ is a Hilbert $N$-bimodule if it admits commuting left and right
normal actions. A vector $\xi\in\mathcal{H}$ is tracial if $\langle
x\xi,\xi\rangle=\langle\xi x,\xi\rangle=\tau(x)$ for all $x\in N$, and it is
$B$-central if $b\xi=\xi b$ for all $b\in B$. A pair $(\mathcal{H},\xi)$ where
$\xi\in\mathcal{H}$ is a unit vector is called a pointed Hilbert $N$-bimodule.

The notion of rigidity of inclusions is due to Popa ([Po06], Sections 4 and
5); however, for practical reasons, we recall Definition 1.3.1 of [Io10]:

###### Definition 3.10

The inclusion $B\subset N$ is rigid (or has relative property (T)) if, for
every $\varepsilon>0$, there exists a finite set $F\subset N$ and $\delta>0$
such that, if $(\mathcal{H},\xi)$ is a pointed Hilbert $N$-bimodule where
$\xi$ is a tracial vector verifying

$\max_{x\in F}\|x\xi-\xi x\|\leq\delta,$

then there exists a $B$-central vector $\eta\in\mathcal{H}$ with
$\|\eta-\xi\|\leq\varepsilon$.

###### Remark 3.11

It is worth being mentioned that Definition 3.10 is in fact independent of the
chosen trace on $N$. Furthermore, if $B\subset N$ is a Cartan subalgebra of
$N$, it follows from Corollary 1.3.3 of [Io10] that the required control on
the distance between the $B$-central vector $\eta$ and the tracial vector
$\xi$ can be omitted: the inclusion is rigid if and only if there exist
$F\subset N$ finite and $\delta>0$ such that any pointed Hilbert $N$-bimodule
$(\mathcal{H},\xi)$, where $\xi$ is tracial and $\|x\xi-\xi x\|\leq\delta$ for
every $x\in F$, contains a non-zero $B$-central vector.

Important examples of rigid inclusions are provided by discrete groups:

###### Proposition 3.12

(Corollary 5.2, [Po06]) Let $G$ be a countable group and let $H\subset G$ be a
subgroup of $G$. Denote by $L(H)\subset L(G)$ their associated von Neumann
algebras. Then the inclusion $L(H)\subset L(G)$ is rigid if and only if the
pair $(G,H)$ has the relative property (T). $\blacksquare$

Let us now recall Ioana’s definition of rigid equivalence relations:

###### Definition 3.13

(Definition 1.3.4, [Io10]) Let $(X,\mu)$ be a probability space and let
$\mathcal{R}$ be a countable, ergodic, measure preserving equivalence relation
on $X$. Then $\mathcal{R}$ is rigid if its associated inclusion
$L^{\infty}(X)\subset L(\mathcal{R})$ is rigid. Also, we say that an
essentially free, ergodic, p.m.p. action $\Gamma\curvearrowright(X,\mu)$ is
rigid if the inclusion $L^{\infty}(X)\subset L^{\infty}(X)\rtimes\Gamma$ is
rigid. Equivalently, the action $\Gamma\curvearrowright(X,\mu)$ is rigid if
and only if the associated equivalence relation $\mathcal{R}_{\Gamma}$ is
rigid.

###### Definition 3.14

Let $\Gamma\curvearrowright(X,\mu)$ be an ergodic and free p.m.p action. Then
the action satisfies Ioana’s dichotomy if every ergodic subequivalence
relation $\mathcal{R}\subset\mathcal{R}_{\Gamma}$ is either amenable (in the
sense of [CFW81]) or rigid.

One of the main goals of the present article is to provide examples of
Haagerup maximal subalgebras that generalize Theorem 3.1 of [JS21], as stated
in Theorem 1.5. Thus, let us consider the following more general setting: let
$\Gamma\curvearrowright(X,\mu)$ be an ergodic and essentially free action so
that $A=L^{\infty}(X)\subset N=A\rtimes\Gamma$ is the inclusion of the Cartan
subalgebra $A$ in the type II1 factor $N$. Assume furthermore that the action
satisfies Ioana’s dichotomy, and let $A\subset P$ be a subfactor which is
maximal amenable in $N$. We intend to prove that $P$ is a maximal Haagerup
subalgebra of $N$. In order to prove this, we consider an intermediate von
Neumann algebra $P\subset Q\subset N$ which is assumed to have the Haagerup
property, and we have to prove that $Q=P$. As a first step, as $A$ is maximal
abelian in $N$, we observe that

$Q^{\prime}\cap N\subset P^{\prime}\cap N\subset A^{\prime}\cap N=A\subset P,$

and as $P$ is a factor, the same holds for $Q$. Hence, by Theorem 1 of
[FM77II], there exists an ergodic equivalence relation $\mathcal{R}$ on $X$
and a Borel $2$-cocycle $s$ on the relation $\mathcal{R}$ such that
$Q=L(\mathcal{R},s)$, as defined after Lemma 3.8 444The authors of [JS21]
appear to have overlooked the possibility that $s$ is non-trivial in the proof
of their Theorem 3.1.. In order to prove the equality $Q=P$, we need to prove
that $\mathcal{R}$ is amenable, using Ioana’s criterion of rigidity for
equivalence relations as in Section 2 of [Io10], but after having proved that
the potential presence of the $2$-cocycle $s$ has no effect on the conclusion.
In other words, we need to adapt Lemma 2.1 and Proposition 2.2 of [Io10].

In order to do that, we denote by $B(X)$ the algebra of all complex-valued,
bounded Borel functions on $X$. For two functions
$f_{1},f_{2}:X\rightarrow\mathbb{C}$, the function $f_{1}\otimes f_{2}:X\times
X\rightarrow\mathbb{C}$ is defined by $(f_{1}\otimes
f_{2})(x_{1},x_{2})=f_{1}(x_{1})f_{2}(x_{2})$. Given two (regular) Borel
probability measures $\mu$ and $\nu$ on $X$, the norm $\|\mu-\nu\|$ is defined
by

$\|\mu-\nu\|=\sup\Big{\\{}\Big{|}\int_{X}fd\mu-\int_{X}fd\nu\Big{|}\colon f\in
B(X),\|f\|_{\infty}\leq 1\Big{\\}}.$

Finally, observe that the quotient map $B(X)\rightarrow L^{\infty}(X,\mu)$
makes any $L^{\infty}(X,\mu)$-bimodule a $B(X)$-bimodule.

###### Lemma 3.15

Let $\mathcal{R}$ be a standard equivalence relation as above on $(X,\mu)$ so
that $\mu$ is invariant under $\mathcal{R}$, and let $s$ be a Borel
$2$-cocycle as above. Let $(\mathcal{H},\xi)$ be a pointed Hilbert
$L(\mathcal{R},s)$-bimodule. Then there exists a probability measure $\nu$ on
$X\times X$ such that

  1. (i)

$\int_{X\times X}(f_{1}\otimes f_{2})d\nu=\langle f_{1}\xi f_{2},\xi\rangle$,
$\forall f_{1},f_{2}\in B(X)$.

  2. (ii)

$\|(\theta\otimes\theta)_{*}\nu-\nu\|\leq 2\|u_{\theta}\xi-\xi u_{\theta}\|$,
$\forall\theta\in[\mathcal{R}]$.

  3. (iii)

If $\xi$ is tracial, then $p_{*}^{i}\nu=\mu$ pour $i=1,2$.

  4. (iv)

If $\mathcal{H}$ has no non-zero $L^{\infty}(X,\mu)$-central vector, then
$\nu(\Delta)=0$.

Proof: The proof is the same as that of Lemma 2.1 in [Io10] since the only
place where $s$ occurs is in the definition of the unitary $u_{\theta}$ and,
due to the latter, is used just in relation (C) of Lemma 3.8, namely
$u_{\theta}au_{\theta}^{*}=a\circ\theta^{-1}$ for $a\in L^{\infty}(X)$.
$\blacksquare$

###### Proposition 3.16

Let $\mathcal{R}$ be as in Lemma 3.15. Suppose moreover that $\mathcal{R}$ is
ergodic. Assume that there is no sequence of probability measures
$(\nu_{k})_{k\geq 1}$ on $X\times X$ such that $\nu_{k}(\Delta)=0$ and
$p_{*}^{i}\nu_{k}=\mu$ for all $i=1,2$ and $k$,

  1. (i)

$\lim_{k\to\infty}\int_{X\times X}(f_{1}\otimes
f_{2})d\nu_{k}=\int_{X}f_{1}f_{2}d\mu$, $\forall f_{1},f_{2}\in B(X)$, and

  2. (ii)

$\lim_{k\to\infty}\|(\theta\otimes\theta)_{*}\nu_{k}-\nu_{k}\|=0$,
$\forall\theta\in[\mathcal{R}]$.

Then, for every Borel $2$-cocycle $s$, the inclusion $L^{\infty}(X,\mu)\subset
L(\mathcal{R},s)$ is rigid.

Proof: Assume that there exists a Borel $2$-cocycle such that the inclusion
$L^{\infty}(X,\mu)\subset L(\mathcal{R},s)$ is not rigid. Using Definition
3.10 and Remark 3.11, we can find a sequence $(\mathcal{H}_{k},\xi_{k})$ of
pointed Hilbert $L(\mathcal{R},s)$-bimodules such that
$\lim_{k\to\infty}\|x\xi_{k}-\xi_{k}x\|=0$ for all $x\in L(\mathcal{R},s)$,
$\xi_{k}$ is tracial, and $\mathcal{H}_{k}$ has no non-zero
$L^{\infty}(X,\mu)$-central vector, for all $k\geq 1$. Let then $\nu_{k}$ be
the probability measure associated to $(\mathcal{H}_{k},\xi_{k})$ by Lemma
3.15. Then we conclude exactly as in the proof of Proposition 2.2 in [Io10].
$\blacksquare$

###  3.4 Proof of Theorem 1.3

In analogy with Theorem 3.1 of [Io10], we will prove:

###### Theorem 3.17

For $n\geq 2$, let $\mathcal{S}_{n}$ denote the orbital equivalence relation
induced by the linear action on $\mathbb{T}^{n}$. Let $\mathcal{R}$ be an
ergodic subequivalence relation of $\mathcal{S}_{n}$. If $\mathcal{R}$ is not
amenable, then there does not exist a sequence of probability measures
$(\nu_{k})_{k\geq 1}$ on $\mathbb{T}^{n}\times\mathbb{T}^{n}$ such that
$\nu_{k}(\Delta)=0$ for all $k$ and

  1. (i)

$\lim_{k\rightarrow\infty}\int_{\mathbb{T}^{n}\rtimes\mathbb{T}^{n}}(f_{1}\otimes
f_{2})\,d\nu_{k}=\int_{\mathbb{T}^{n}}f_{1}f_{2}\,d\lambda_{n}$ for every
$f_{1},f_{2}\in B(\mathbb{T}^{n})$; here $d\lambda_{n}$ denotes the normalized
Lebesgue measure on $\mathbb{T}^{n}$;

  2. (ii)

$\lim_{k\rightarrow\infty}\Big{|}\int_{\mathbb{T}^{n}\times\mathbb{T}^{n}}(f\circ(\theta\times\theta))\,d\nu_{k}-\int_{\mathbb{T}^{n}\times\mathbb{T}^{n}}f\,d\nu_{k}\Big{|}=0$
for every $f\in B(\mathbb{T}^{n}\times\mathbb{T}^{n})$ and
$\theta\in[\mathcal{R}]$.

The following consequence follows immediately by combining Theorem 3.17 with
Proposition 3.16; it strengthens Theorem 1.3.

###### Corollary 3.18

For $n\geq 2$, let $\mathcal{R}$ be an ergodic sub-equivalence relation of
$\mathcal{S}_{n}$. Then either $\mathcal{R}$ is amenable or, for every Borel
$2$-cocycle $s$, the inclusion $L^{\infty}(X,\mu)\subset L(\mathcal{R},s)$ is
rigid. $\blacksquare$

Proof of Theorem 3.17: Working by contrapositive, we assume that there exist
probability measures $\nu_{k}$ on $\mathbb{T}^{n}\times\mathbb{T}^{n}$, with
$\nu_{k}(\Delta)=0$ for every $k$, and satisfying (i) and (ii), and our aim is
to prove that $\mathcal{R}$ is amenable. Now the proof is copying the 4 steps
of the proof of Theorem 3.1 in [Io10], replacing the
$\operatorname{SL}_{2}(\mathbb{Z})$-action on $\mathbb{T}^{2}$ by the
$G_{n}$-action on $\mathbb{T}^{n}$, so we will merely indicate the non-obvious
changes.

Step 1 goes through.

Notations: We replace Ioana’s maps $\sigma,r,\chi,q,p,\pi$ by the following
ones:

  * •

$\sigma:\mathbb{R}^{n}\setminus\\{0\\}\rightarrow\mathbb{P}^{n-1}(\mathbb{R})$
is the canonical map, and
$r=id\times\sigma:\mathbb{T}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})\rightarrow\mathbb{T}^{n}\times\mathbb{P}^{n-1}(\mathbb{R})$.

  * •

$\chi:[-\frac{1}{2},\frac{1}{2}[^{n}\rightarrow\mathbb{T}^{n}$ is the
continuous bijection given by $\chi(x_{1},\ldots,x_{n})=(e^{2\pi
ix_{1}},\ldots,e^{2\pi ix_{n}})$; let
$\rho:\mathbb{T}^{n}\setminus\\{(1,\ldots,1)\\}\rightarrow\mathbb{R}^{n}\setminus\\{0\\}$
denote the inverse of $\chi$ restricted to
$\mathbb{T}^{n}\setminus\\{(1,\ldots,1)\\}$. Set
$q=id\times\rho:\mathbb{T}^{n}\times(\mathbb{T}^{n}\setminus\\{(1,\ldots,1)\\})\rightarrow\mathbb{T}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$.

  * •

$p:(\mathbb{T}^{n}\times\mathbb{T}^{n})\setminus\Delta\rightarrow\mathbb{T}^{n}\times(\mathbb{T}^{n}\setminus\\{(1,\ldots,1)\\})$
is the homeomorphism given by $p(x,y)=(x,x^{-1}y)$.

  * •

Finally

$\pi=r\circ q\circ
p:(\mathbb{T}^{n}\times\mathbb{T}^{n})\setminus\Delta\rightarrow\mathbb{T}^{n}\times\mathbb{P}^{n-1}(\mathbb{R}):(x,y)\to(x,(\sigma\circ\rho)(x^{-1}y)).$

Remark 3.3 of [Io10], on $G_{n}$-equivariance of $\sigma,r$ and $p$, and
limited equivariance of $\rho$, then goes through without change.

Steps 2 and 3 go through without further change.

Step 4: We must replace the amenability of the
$\operatorname{SL}_{2}(\mathbb{Z})$-action on $\mathbb{P}^{1}(\mathbb{R})$ by
the amenability of the $G_{n}$-action on $\mathbb{P}^{n-1}(\mathbb{R})$, which
follows from Proposition 3.6 together with the fact that amenability of
actions is preserved by closed subgroups (see Proposition 4.3.2 in [Zi84]).
Once this is done, the rest of the proof of Step 4 in [Io10] proceeds without
change. $\blacksquare$

##  4 Von Neumann algebraic applications

###  4.1 Relative property H and HT Cartan subalgebras

As is Section 3, let $N$ be a finite von Neumann algebra with separable
predual equipped with a trace $\tau$. For a von Neumann subalgebra $B\subset
N$, let $E_{B}$ be the unique $\tau$-preserving conditional expectation from
$N$ onto $B$. It extends to the orthogonal projection from $L^{2}(N,\tau)$
onto $L^{2}(B,\tau)$. More generally, if $\phi:N\rightarrow N$ is a completely
positive map such that $\tau\circ\phi\leq\tau$, then it is normal and it
extends to a bounded operator $T_{\phi}$ on $L^{2}(N,\tau)$ characterized by
$T_{\phi}(\hat{x})=\widehat{\phi(x)}$ for every $x\in N$. Notice that $\phi$
is $B$-bimodular (i.e. $\phi(b_{1}xb_{2})=b_{1}\phi(x)b_{2}$ for all
$b_{1},b_{2}\in B$ and $x\in N$) if and only if $T_{\phi}\in B^{\prime}\cap
JB^{\prime}J$. The von Neumann algebra $JB^{\prime}J$ coincides with the so-
called basic construction $\langle N,e_{B}\rangle$ which is generated by $N$
and the projection $e_{B}$ (see for instance Subsection 1.3.1 of [Po06]). As
its commutant $JBJ$ is a finite von Neumann algebra, $\langle N,e_{B}\rangle$
is semifinite; we then denote by $\mathcal{J}(\langle N,e_{B}\rangle)$ the
norm-closed, two-sided ideal in $\langle N,e_{B}\rangle$ generated by its
finite projections. If $B=\mathbb{C}1$, then, as $\langle
N,e_{B}\rangle=JB^{\prime}J$ coincides with the algebra of all bounded
operators on $L^{2}(N,\tau)$, the ideal $\mathcal{J}(\langle N,e_{B}\rangle)$
is the ideal of all compact operators $\mathcal{K}(L^{2}(N,\tau))$.

The following definition is due to Popa [Po06]; notice that the Haagerup
property considered in [Jo02] corresponds to the case $B=\mathbb{C}1$.

###### Definition 4.1

Let $B\subset N$ be von Neumann algebras as above. Then $N$ has property H
relative to $B$ if there exists a sequence $(\phi_{n})$ of normal, completely
positive $B$-bimodular maps on $N$ satisfying the following conditions:

  1. (1)

$\phi_{n}(1)\leq 1$ and $\tau\circ\phi_{n}\leq\tau$ for every $n$;

  2. (2)

$(T_{\phi_{n}})\subset\mathcal{J}(\langle N,e_{B}\rangle)$;

  3. (3)

$\lim_{n\to\infty}\|\phi_{n}(x)-x\|_{2}=0$ for every $x\in N$.

Typical and important examples of inclusions of finite von Neumann algebras
with property H as above come from crossed products: assume that a countable
group $G$ acts on some finite von Neumann algebra $B$ and that the action,
denoted by $\sigma$, preserves some trace $\tau$.

Then one has:

###### Proposition 4.2

([Po06], Proposition 3.1) Let $G$, $(B,\tau)$ and $\sigma$ be as above. Then
$N=B\rtimes_{\sigma}G$ has property H relative to $B$ if and only if the group
$G$ has the Haagerup property. $\blacksquare$

Now, let us consider the following situation: let $\Gamma$ be a countable
group that acts by automorphisms on some discrete group $H$. Then the von
Neumann algebra $L(H\rtimes\Gamma)$ associated to the semidirect product group
$G=H\rtimes\Gamma$ is naturally isomorphic to the crossed product algebra
$N=L(H)\rtimes\Gamma$. Assume that $\Gamma$ has the Haagerup property and that
the pair $(G,H)$ has the relative property (T). Then $L(H)$ is an HT
subalgebra of $N$ in the sense of the following definition:

###### Definition 4.3

(Definition 6.1, [Po06]) Let $N$ be a finite von Neumann algebra and let
$B\subset N$ be a von Neumann subalgebra. Then $B$ is an HT subalgebra of $N$
if the following two conditions are met:

  1. (a)

$N$ has property H relative to $B$;

  2. (b)

there exists a von Neumann subalgebra $B_{0}\subset B$ such that
$B^{\prime}_{0}\cap N\subset B$ and the inclusion $B_{0}\subset N$ is rigid.

If $A\subset N$ is a Cartan subalgebra that satisfies both conditions, then it
is called an HT Cartan subalgebra.

The importance of existence of HT Cartan subalgebras is illustrated by the
following results:

###### Theorem 4.4

(Theorem 6.2, [Po06]) Let $N$ be a type II1 factor with two HT Cartan
subalgebras $A_{1}$ and $A_{2}$. Then there exists a unitary element $u\in N$
such that $uA_{1}u^{*}=A_{2}$. $\blacksquare$

###### Corollary 4.5

(Corollaries 6.5 and 8.2, [Po06]) Let $A_{i}\subset N_{i}$, $i=1,2$, be HT
Cartan subalgebras of type II1 factors, and let $\theta:N_{1}\rightarrow
N_{2}$ be an isomorphism. Then there exists $u\in U(N_{2})$ such that
$u\theta(A_{1})u^{*}=A_{2}$. Thus, if $A\subset N$ is an HT Cartan subalgebra
of a type II1 factor $N$, there exists a unique (up to isomorphism) standard
equivalence relation $\mathcal{R}_{N}^{HT}$ on the standard probability space
$(X,\mu)$ such that the inclusion $A\subset N$ is isomorphic to the inclusion
$L^{\infty}(X,\mu)\subset L(\mathcal{R}_{N}^{HT},s)$ for some cocycle $s$.
Thus the factor $N$ inherits $\ell^{2}$-Betti numbers
$\beta_{k}^{HT}(N)\coloneqq\beta_{k}(\mathcal{R}_{N}^{HT})$ for every integer
$k\geq 0$, which are isomorphism invariants of $N$. $\blacksquare$

Following Notation 6.4 of [Po06], we denote by $\mathcal{HT}$ the class of all
type II1 factors with separable predual with HT Cartan subalgebras.

Another interesting property of the $\ell^{2}$-Betti numbers
$(\beta_{k}^{HT}(N))_{k\geq 0}$ for $N\in\mathcal{HT}$ is that, for every real
number $t>0$, we have $\beta_{k}^{HT}(N^{t})=\beta_{k}^{HT}(N)/t$ (Corollary
8.2, [Po06]), where $N^{t}$ denotes the amplification of $N$ by $t$, which is
the factor $pM_{n}(N)p$, where $n\geq t$ is a positive integer and $p\in
M_{n}(N)$ is a projection with (non-normalized) trace
$\operatorname{Tr}\otimes\tau(p)=t$. (We recall that the isomorphism class of
$N^{t}$ depends neither on $p$ nor on $n\geq t$.)

Recall also that if $N$ is a type II1 factor, its fundamental group
$\mathcal{F}(N)$ is the set of all $t>0$ such that $N^{t}$ is isomorphic to
$N$. Hence, if $N\in\mathcal{HT}$ is such that $0<\beta_{k}(N)<+\infty$ for
some $k$, then $\mathcal{F}(N)$ is trivial.

Proof of Theorem 1.4: Set $A_{n}=L^{\infty}(\mathbb{T}^{n})$ which is a Cartan
subalgebra of $N_{n}$. Then, by Proposition 3.1 of [Va23], the inclusion
$A_{n}\subset N_{n}$ is rigid, and as $G_{n}$ has the Haagerup property,
$A_{n}$ is an HT Cartan subalgebra of $N_{n}$. Furthermore, it follows from
Corollary 3.16 in [Ga00], that
$\beta_{k}(\mathcal{R}_{N}^{HT})=\beta_{k}(G_{n})$ for all $k\geq 0$ and
$n\geq 2$. The rest is a consequence of the above discussion and Corollary
4.5. $\blacksquare$

###  4.2 Proof of Theorem 1.5

First, we need to prove the existence of maximal Haagerup von Neumann algebras
in finite ones.

In fact, we will prove a more general result which not only yields the above
statement, but which implies the following result of independent interest:
given a finite von Neumann algebra $(N,\tau)$ with separable predual, the
subset $\operatorname{HAP}(N)$ of all von Neumann subalgebras of $N$ that have
the Haagerup property is closed with respect to the Effros-Maréchal topology
that we recall now (see for instance [HW98] or [AHO23]). Denote by
$\operatorname{SA}(N)$ the set of all von Neumann subalgebras of $N$. As the
latter is finite, the topology on $\operatorname{SA}(N)$ can be described as
follows: a net $(M_{i})_{i\in I}\subset\operatorname{SA}(N)$ converges to
$M\subset N$ if and only

$\liminf_{i\in I}M_{i}=\limsup_{i\in I}M_{i}=M,$

where

$\liminf_{i\in I}M_{i}\coloneqq\\{x\in N\colon\exists(x_{i})_{i\in
I}\in\ell^{\infty}(I,M_{i})\ :\ \lim_{i\in I}\|x_{i}-x\|_{2}=0\\}$

and

$\limsup_{i\in I}M_{i}\coloneqq\langle\\{x\in N\colon\exists(x_{i})_{i\in
I}\in\ell^{2}(I,M_{i})\ :\ \mathrm{wo}-\lim_{i\in I}x_{i}=x\\}\rangle.$

Notation $\langle E\rangle$ means the von Neumann algebra generated by the
subset $E\subset N$, wo-$\lim$ refers to the limit with respect to the weak
operator topology on $N$ and $\ell^{\infty}(I,M_{i})$ denotes the C∗-algebra
of bounded nets $(x_{i})_{i\in I}\in\prod_{i}M_{i}$.

Statement (1) of the next result555It was inspired by Proposition 4.7 in
[AHO23] which states that the subset of amenable von Neumann subalgebras of
$N$ is closed. is a von Neumann algebraic analogue of the fact that, for a
countable group $G$, the set of Haagerup subgroups is closed in the Chabauty
topology on the set of all subgroups of $G$ (which follows easily from
Proposition 6.1.1 in [CCJJV01]).

###### Proposition 4.6

Let $(N,\tau)$ be a finite von Neumann algebra as above and let $(M_{i})_{i\in
I}\subset\operatorname{HAP}(N)$ be a net such that $M\coloneqq\liminf_{i\in
I}M_{i}$ exists. Then $M\in\operatorname{HAP}(N)$. In particular,

  1. (1)

the subset $\operatorname{HAP}(N)$ is closed in $\operatorname{SA}(N)$ with
respect to the Effros-Maréchal topology and,

  2. (2)

for every von Neumann subalgebra $P\in\operatorname{HAP}(N)$ , there exists a
maximal Haagerup von Neumann subalgebra $Q\subset N$ which contains $P$.

Proof: Denoting by $(N)_{1}$ the unit ball of $N$ for the operator norm, and
setting $(E)_{1}=E\cap(N)_{1}$ for every subset $E$ of $N$, let $(M_{i})_{i\in
I}\in\operatorname{HAP}(N)$ and $M\in SA(N)$ be such that $M=\liminf_{i\in
I}M_{i}$.
To show that $M\in\operatorname{HAP}(N)$, it suffices to prove that, for every
finite set $F\subset(M)_{1}$ and every $\varepsilon>0$, there exists a
completely positive map $\Phi=\Phi_{(F,\varepsilon)}:M\rightarrow M$ with the
following properties:

  1. (a)

$\Phi(1)\leq 1$ and $\tau\circ\Phi\leq\tau$;

  2. (b)

the extension $T_{\Phi}$ of $\Phi$ on $L^{2}(M,\tau)$ is a compact operator;

  3. (c)

$\max_{x\in F}\|\Phi(x)-x\|_{2}\leq\varepsilon$.

Thus, let $F\subset(Q)_{1}$ and $\varepsilon>0$ as above. Let us number the
elements of $F$ so that $F=\\{x_{1},\ldots,x_{r}\\}$. For each $1\leq k\leq
r$, there exists a net $(x_{k}^{(i)})_{i\in I}\in\ell^{\infty}(I,M_{i})$ such
that $x_{k}^{(i)}\in(M_{i})_{1}$ for every $i$ and that

$\lim_{i\in I}\|x_{k}^{(i)}-x_{k}\|_{2}=0.$

Fix henceforth an element $j\in I$ such that

$\max_{1\leq k\leq r}\|x_{k}^{(j)}-x_{k}\|_{2}\leq\varepsilon/3.$

Since $M_{j}\in\operatorname{HAP}(N)$, there exists a completely positive map
$\phi_{j}:M_{j}\rightarrow M_{j}$ which has properties (a) and (b), and such
that

$\max_{1\leq k\leq
r}\|\phi_{j}(x_{k}^{(j)})-x_{k}^{(j)}\|_{2}\leq\varepsilon/3.$

Set $\Phi\coloneqq E_{M}\circ\phi_{j}\circ E_{M_{j}}|_{M}:M\rightarrow M$. It
obviously satisfies condition (a) above. Moreover,
$T_{\Phi}=e_{M}T_{\phi_{j}}e_{M_{j}}|_{L^{2}(M,\tau)}$ is a compact operator
on $L^{2}(M,\tau)$, and finally we have for every $1\leq k\leq r$, since
$x_{k}\in M$,

$\displaystyle\|\Phi(x_{k})-x_{k}\|_{2}$
$\displaystyle=\|E_{M}[\phi_{j}(E_{M_{j}}(x_{k}))-x_{k}]\|_{2}\leq\|\phi_{j}(E_{M_{j}}(x_{k}))-x_{k}\|_{2}$
$\displaystyle\leq\|\phi_{j}[E_{M_{j}}(x_{k})-x_{k}^{(j)}]\|_{2}+\|\phi_{j}(x_{k}^{(j)})-x_{k}^{(j)}\|_{2}+\|x_{k}^{(j)}-x_{k}\|_{2}$
$\displaystyle\leq
2\|x_{k}^{(j)}-x_{k}\|_{2}+\|\phi_{i}(x_{k}^{(j)})-x_{k}^{(j)}\|_{2}\leq\varepsilon.$

(We have used the inequality $\|\phi_{j}(y)\|_{2}\leq\|y\|_{2}$ for every
$y\in M_{j}$ which follows from $\tau\circ\phi_{j}\leq\tau$.)
Statement (1) follows then trivially, and, in order to prove Statement (2), by
Zorn’s lemma, it suffices to prove that, if $(Q_{i})_{i\in I}$ is an
increasing net in $\operatorname{HAP}(N)$ such that $P\subset Q_{i}$ for every
$i$, then $Q\coloneqq(\bigcup_{i\in
I}Q_{i})^{\prime\prime}\in\operatorname{HAP}(N)$. But obviously
$Q=\liminf_{i}Q_{i}$, and $Q\in\operatorname{HAP}(N)$ by the main part of the
proof. $\blacksquare$

Proof of Theorem 1.5: Fix $n\geq 2$. The abelian subalgebra
$A_{n}=L^{\infty}(\mathbb{T}^{n})$ being a Cartan subalgebra of $N_{n}$, by
Proposition 3.6 of [Po81], $A_{n}$ is contained in an intermediate hyperfinite
subfactor $P$ of $N_{n}$ in which it is still a Cartan subalgebra. Let then
$P\subset Q\subset N_{n}$ be a maximal Haagerup von Neumann subalgebra whose
existence is guaranteed by Lemma 4.6. By the considerations following
Definition 3.14, we see that $Q$ is a factor, hence $Q=L(\mathcal{R},s)$,
where $\mathcal{R}$ is an ergodic subequivalence relation of $\mathcal{S}_{n}$
and $s$ is a Borel 2-cocycle on $\mathbb{R}$. By Corollary 3.18, either
$\mathcal{R}$ is amenable or the inclusion $L^{\infty}(X,\mu)\subset
L(\mathcal{R},s)=Q$ is rigid. The second case is precluded by the assumption
that $Q$ has the Haagerup property, so $\mathcal{R}$ is amenable, hence $s$ is
trivial and $Q$ is an hyperfinite von Neumann subalgebra.

Next, fix $n\geq 2$ even. Let $H$ be a maximal amenable subgroup of
$\operatorname{SL}_{2}(\mathbb{Z})$ containing some hyperbolic matrix. Set
$P=A_{n}\rtimes H$; view $P$ as a von Neumann subalgebra of $N_{n}$. Then $P$
is a subfactor, by the second part of Theorem 1.1. Moreover $P$ is maximal
amenable in $N_{n}$, by a result of Boutonnet and Carderi (Theorem A and
Corollary B in [BC15]). The above argument shows that $P$ is maximal Haagerup
in $N_{n}$.

Finally, if $n\geq 3$ is odd, and if $P$ and $Q$ are as in the first part of
the proof, being factors, they cannot be of the form $L(K)$ for any subgroup
$K$ of $\mathbb{Z}^{n}\rtimes\operatorname{PSL}_{2}(\mathbb{Z})$ by
Proposition 1.2. $\blacksquare$

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Authors addresses:

Institut de mathématiques
Université de Neuchâtel
11 Rue Emile Argand - Unimail
CH-2000 Neuchâtel - SUISSE

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