# On relative commutants of subalgebras in group and tracial crossed product
von Neumann algebras

Tattwamasi Amrutam  Ben Gurion University of the Negev. Department of
Mathematics. Be’er Sheva, 8410501, Israel<EMAIL_ADDRESS>and
Jacopo Bassi<EMAIL_ADDRESS>
###### Abstract.

Let $\Gamma$ be a discrete group acting on a compact Hausdorff space $X$.
Given $x\in X$, and $\mu\in\text{Prob}(X)$, we introduce the notion of
contraction of $\mu$ towards $x$ with respect to unitary elements of a group
von Neumann algebra not necessarily coming from group elements. Using this
notion, we study relative commutants of subalgebras in tracial crossed product
von Neumann algebras. The results are applied to negatively curved groups and
$\operatorname{SL}(d,\mathbb{Z})$, $d\geq 2$.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the paper
    2. 1.2 Acknowledgement
  2. 2 Preliminaries and Technicalities
    1. 2.1 Group von Neumann algebra
  3. 3 Crossed product of tracial von Neumann algebras
  4. 4 Relative commutants of subgroups of negatively curved groups
  5. 5 The case of $\operatorname{SL}(d,\mathbb{\mathbb{Z}})$

##  1\. Introduction

Operator algebras associated with discrete groups, or more generally discrete
group actions, reveal essential properties of the underlying group. Probably
the first evidence of this connection is that amenability has neat
characterizations at the operator-algebraic level: injectivity of the group
von Neumann algebra and nuclearity of the reduced group $C^{*}$-algebra.
Nowadays, it is known that many other group properties have analog
descriptions in terms of group $C^{*}$-algebras, for example, a-T-menability
and property T ([20]). Also, free groups’ full and reduced $C^{*}$-algebras
can detect their order. It is a significant open problem raised by A. Connes
whether non-isomorphic ICC property T groups have non-isomorphic von Neumann-
algebras.

Dynamical systems represent a powerful tool for the study of rigidity
properties of groups. As an example, rigidity results for certain discrete
subgroups of $\operatorname{SL}(2,\mathbb{R})$ can be obtained by looking at
the $C^{*}$-crossed products associated with certain actions (cf. [9, 10,
27]). Among the possible dynamical systems, an important role is played by
boundary actions; for example, the topological amenability of the left action
of a discrete group $\Gamma$ on its Stone-Čech boundary
$\partial_{\beta}\Gamma$ is equivalent to exactness and topological
amenability of the left-right action on the same space (which is usually
referred to as bi-exactness or property $\mathcal{S}$) implies the Akemann-
Ostrand (AO) property ([4]), temperateness of the representation of
$\Gamma\times\Gamma$ on the Calkin algebra of $l^{2}(\Gamma))$, which ensures
solidity (hence primeness) of the group von Neumann algebra. A significant
open problem in the theory is deciding whether these three properties coincide
(cf. [11]). The solidity of the group von Neumann algebra is a very rigid
property, which, for example, captures to some extent the dimension of the
ambient group in the case of a lattice in a simple Lie group: it is automatic
for discrete subgroups of simple rank-1 Lie groups, and it is automatically
denied by the existence of infinite subgroups with non-amenable centralizers.
Weakenings of the (AO) property have been considered in the literature and
lead to the definition of properly proximal groups ([18]), for which some
weaker rigidity properties hold as well.

One of the most significant breakthroughs of recent years is the recognition
of the central role of proximality arguments in the study of rigidity
properties of discrete groups through the lens of dynamical systems, which
lead, for example, to the identification of the Furstenberg boundary of a
discrete group with the equivariant Hamana-injective envelope of the complex
numbers (cf., for example, [30, 35, 12]).

Given a probability measure, $\mu\in\text{Prob}(X)$, proximality of $\mu$ is
nothing but the contraction of this measure with a specific sequence of group
elements. In addition, if these group elements leave every finite subset of
the group $\Gamma$, then it can be shown that the corresponding unitary
elements converge to zero weakly. We generalize this notion to the context of
general unitary elements inside the group von Neumann algebra, which does not
necessarily come from group elements.

######  Definition 1.1.

Let $\Gamma$ be a discrete countable group acting on a compact Hausdorff space
$X$ endowed with a probability measure $\mu$. Let $(u_{n})$ be a sequence of
unitaries in $L\Gamma$. We say that $u_{n}$ is $\mu$-contracting (towards a
point $x\in X$) if for every $\epsilon>0$, every $F\subset\Gamma$ finite and
for every $f\in C(X)$ there is $N$ such that for every $n\geq N$ we have
$\|u_{n}|_{A}\|_{2}>1-\epsilon$, where
$A=\\{\gamma\in\Gamma\;|\;|\gamma^{-1}\eta\mu(f)-f(x)|<\epsilon\
\;\forall\eta\in F\\}$.

If $u_{n}$’s come from the group elements, then the notion of
$\mu$-contraction agrees with that of the $\Gamma$-contraction. Motivated by
the notion of solid von Neumann algebra, we employ proximality arguments to
study the position of relative commutants of subalgebras in group (and more
generally crossed product) von Neumann algebras.

######  Theorem 1.2.

Let $\Gamma$ be a discrete countable group acting on a compact Hausdorff space
$X$. Let $\\{u_{n}\\}\subset L(\Gamma)$ be a $\mu$-contracting sequence for
the action on $X$ for some probability measure $\mu$ on $X$. Let
$(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann algebra, and
$\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a trace-preserving
action. Then
$\\{u_{n}:n\in\mathbb{N}\\}^{\prime}\cap(\mathcal{N}\rtimes\Gamma)\subset\mathcal{N}\rtimes\Gamma_{x}$,
where $x$ is determined by the fact that $(u_{n})$ is $\mu$-contracting
towards $x$, and $\Gamma_{x}=\\{s\in\Gamma:sx=x\\}$.

###  1.1. Organization of the paper

Apart from this section, there are four other sections. We prove some
preliminary technicalities in Section 2. In Section 3, using a suitable notion
of convergence of a measure to a point under a sequence of unitaries, inspired
by [18], we show that the commutant of certain subalgebras of tracial crossed
product algebras is contained in the von Neumann algebra associated to the
stabilizer of the limiting point. In Section 4, we consider the case of
"negatively curved groups" and show that in this case, the position of the
relative commutants of subgroup algebras is reminiscent of an averaging
property, in the spirit of the Powers’ averaging property (cf. for example
[37, 3, 36]). In Section 5, we consider the case of infinite subgroups of
$\operatorname{SL}(d,\mathbb{Z})$, in which case we prove that the position of
the relative commutant of a subgroup algebra depends on the dynamics of the
subgroup in a particular partial flag. Per the results appearing in [14], we
also see that a weak form of solidity holds for
$\operatorname{SL}(3,\mathbb{Z})$.

The authors believe that the techniques developed in this manuscript should
have a deep connection with the approach appearing in [15] and [13]. This
connection will be investigated in future work.

###  1.2. Acknowledgement

The first named author thanks Yair Glasner and Ido Grayevsky for many helpful
discussions around hyperbolic groups. The authors thank Yongle Jiang for
reading through a near complete draft of this paper and for his numerous
suggestions and corrections. The authors are also supported by research
funding from the European Research Council (ERC) under the European Union’s
Seventh Framework Program (FP7-2007-2013) (Grant agreement No. 101078193). The
second named author acknowledges the support of INdAM-GNAMPA and the grant
"Operator Algebras and Quantum Mathematics (OAQM)," CUP: E83C22001800005.

##  2\. Preliminaries and Technicalities

Let $\Gamma$ be a discrete group. By a $\Gamma$-space $X$, (also denoted as
$\Gamma\curvearrowright X$ sometimes) on a compact Hausdorff space $X$, we
mean a group homomorphism $\pi:\Gamma\to\text{Homeo}(X)$. We often abuse the
notation by ignoring $\pi$ and write $sx$ instead of $\pi(s)x$ for
$s\in\Gamma$ and $x\in X$.

######  Definition 2.1.

Let $\Gamma$ be a discrete countable group acting on a second countable
compact Hausdorff space $X$ endowed with a probability measure $\mu$. A
sequence $(\gamma_{n})\subset\Gamma$ is said to be a $\mu$-pointwise
contracting sequence if there is $x\in X$ such that $\gamma_{n}y\rightarrow x$
for $\mu$-almost every $y\in X$. In this case we say that $(\gamma_{n})$ is
$\mu$-pointwise-contracting towards $x$.

If $\mu$ is point-wise contracted by the sequence $\\{\gamma_{n}\\}$, then it
is also pointwise contracted by $\\{\gamma_{n}\gamma\\}$ for any group element
$\gamma\in\Gamma$. Moreover, the point where it converges is also unchanged.
In other words, the contracting sequence is invariant with respect to the
right multiplication by the group elements. We make this precise below.

######  Lemma 2.2.

Let $\Gamma$ be a discrete countable group acting on a second countable
compact space $X$ and $\mu$ a $\Gamma$-quasi invariant probability measure on
$X$. Suppose that there is a $\mu$-pointwise-contracting sequence
$(\gamma_{n})\subset\Gamma$ towards a point $x\in X$. Then for every
$\gamma\in\Gamma$ we have that $\lim_{n}\gamma_{n}\gamma\mu=\delta_{x}$ exists
and is independent of $\gamma$.

###### Proof.

There is $x\in X$ such that for $\mu$-almost every $y\in X$ we have
$\lim_{n}\gamma_{n}y\rightarrow x$. Let $E\subset X$ be the subset of
$\mu$-measure $0$ such that $\lim_{n}\gamma_{n}y=x$ for every $y\notin E$.
Since $\mu$ is quasi invariant we have that for every $\gamma\in\Gamma$ there
is a subset $E^{\prime}$ of measure zero, namely $\gamma^{-1}E$, such that
$\lim_{n}\gamma_{n}\gamma y=x$ for every $y\notin E^{\prime}$. Hence, for
every $f\in C(X)$ and every $\gamma\in\Gamma$ we have $f(\gamma_{n}\gamma
y)\rightarrow f(x)$ $\mu$-almost everywhere. It follows from Lebesgue
dominated convergence Theorem that
$\lim_{n}\gamma_{n}\gamma\mu=\lim_{n}\gamma_{n}\mu$ in the weak∗-topology. ∎

###  2.1. Group von Neumann algebra

We briefly recall the construction of the group von Neumann algebra. Let
$\ell^{2}(\Gamma)$ be the space of square summable $\mathbb{C}$-valued
functions on $\Gamma$. There is a natural action
$\Gamma\curvearrowright\ell^{2}(\Gamma)$ by left translation:

$\lambda_{g}\xi(h):=\xi(g^{-1}h),\xi\in\ell^{2}(\Gamma),g,h\in\Gamma$

The group von Neumann algebra $L(\Gamma)$ is generated (as a von Neumann
algebra inside $\mathbb{B}(\ell^{2}(\Gamma)$), by the left regular
representation $\lambda$ of $\Gamma$. The group von Neumann algebra
$L(\Gamma)$ comes equipped with a canonical trace
$\tau_{0}:L(\Gamma)\to\mathbb{C}$ defined by

$\tau_{0}\left(\lambda_{g}\right)=\left\\{\begin{array}[]{ll}0&\mbox{if $g\neq
e$}\\\ 1&\mbox{if $g=e$}\end{array}\right\\}$

It is worth noting that a natural embedding of $L(\Gamma)$ into
$\ell^{2}(\Gamma)$ exists via the map $x\mapsto x\delta_{e}$. Therefore, any
element $x\in L(\Gamma)$ can be expressed as
$x=\sum_{g\in\Gamma}x_{g}\lambda(g)$, where $\lambda(g)\in L(\Gamma)$
correspond to the canonical unitaries of $L(\Gamma)$ and
$x_{g}=\tau_{0}(x\lambda(g)^{*})$ are the Fourier coefficients of $x$. The
above sum converges in $\ell^{2}$-norm ($\|\cdot\|_{2}$) and not with respect
to the strong operator or weak operator topology, as mentioned in [5, Remark
1.3.7]. This expansion is commonly referred to as the Fourier expansion of
$x$.

We will make use of the following notion, which generalizes Definition 2.1 to
sequences of unitaries in a group von Neumann algebra, which do not come from
group elements.

######  Definition 2.3.

Let $\Gamma$ be a discrete countable group acting on a compact Hausdorff space
$X$ endowed with a probability measure $\mu$. Let $(u_{n})$ be a sequence of
unitaries in $L\Gamma$. We say that $u_{n}$ is $\mu$-contracting (towards a
point $x\in X$) if for every $\epsilon>0$, every $F\subset\Gamma$ finite and
for every $f\in C(X)$ there is $N$ such that for every $n\geq N$ we have
$\|u_{n}|_{A}\|_{2}>1-\epsilon$, where
$A=\\{\gamma\in\Gamma\;|\;|\gamma^{-1}\eta\mu(f)-f(x)|<\epsilon\
\;\forall\eta\in F\\}$.

######  Lemma 2.4.

Let $\Gamma$ be a discrete countable group acting on a second countable
compact space $X$ and $\mu$ a $\Gamma$-quasi invariant probability measure on
$X$. Let $\Lambda\subset\Gamma$ be a subgroup with the property that there is
a point $x\in X$ such that every diverging sequence $\lambda_{n}$ in $\Lambda$
is $\mu$-pointwise-contracting towards $x$. Then every sequence of unitaries
in $L\Lambda$ which goes to zero weakly is $\mu$-contracting towards $x$.

###### Proof.

Let $\epsilon>0$. Since every diverging sequence in $\Lambda$ is
$\mu$-contracting towards $x$, it follows from Lemma 2.2 that for every finite
set $G\subset\Gamma$ and every $f\in C(X)$ there is a finite set
$F\subset\Lambda$ satisfying $|\lambda^{-1}\gamma\mu(f)-f(x)|<\epsilon$ for
every $\lambda\in\Lambda\backslash F$, $\gamma\in G$. Now, since
$u_{n}\rightarrow 0$ weakly, there is $N\in\mathbb{N}$ such that
$\|u_{n}|_{F}\|_{2}<\epsilon$ for every $n>N$. The result follows. ∎

######  Example 2.5.

Let $\Gamma$ be a convergence group, i.e., there exists an action
$\Gamma\curvearrowright X$ such that for every distinct sequence of elements
$\\{g_{n}\\}\subset\Gamma$, we can find two elements $a,b\in X$ and a sub-
sequence $g_{n_{k}}$ such that $g_{n_{k}}|_{X\setminus\\{b\\}}\to a$ uniformly
on every compact subset of $X\setminus\\{b\\}$ (see [39]). In this case, $a$
is called the attracting point, and $b$ is the repelling point. Let
$s\in\Gamma$ be a parabolic element, i.e., $s$ has exactly one fixed point on
$X$. Let us call it $x_{s}^{+}$. Let $\Lambda=\langle s\rangle$. Using [39,
Lemma 2F], we see that $\\{s^{n}\\}_{n\in\mathbb{Z}}$ is a convergence
sequence with the attractive and repelling point of
$\\{s^{n}\\}_{n\in\mathbb{Z}}$ the same as that of $\\{x_{s}^{+}\\}$. Let us
assume that $\Gamma$ is non-elementary, i.e., the set of limit points $LX$
(the collection of all attracting points on $X$) has more than two points.
Using [39, Theorem 2S], we see that $LX$ is an infinite perfect set. Let $\mu$
be a $\Gamma$-quasi invariant probability measure on $X$ such that
$\mu(x_{s}^{+})\neq 0$. It follows from definition 2.1 that every diverging
sequence $\lambda_{n}$ in $\Lambda$ is $\mu$-pointwise-contracting towards
$x_{s}^{+}$. Let $\mathcal{M}\leq L(\Lambda)$ be a diffuse von Neumann
subalgebra. Let $u_{n}\in\mathcal{M}$ be a sequence of unitaries in
$\mathcal{M}$ which converges to $0$ weakly. It follows from Lemma 2.4 that
$u_{n}$ is $\mu$-contracting towards $x_{s}^{+}$. Similarly, if $s\in\Gamma$
is loxodromic, then there is a quasi-invariant probability measure $\mu$ on
$X$ and points $x_{s}^{+}$, $x_{s}^{-}$ such that the sequence $(s^{n})$ is
$\mu$-pointwise-contracting towards $x_{s}^{+}$ and $(s^{-n})$ is
$\mu$-pointwise-contracting towards $x_{s}^{-1}$.

Given a $\Gamma$-space $X$, and $\mu\in\text{Prob}(X)$, the Poisson
transformation $P_{\mu}:C(X)\to\mathbb{B}(\ell^{2}(\Gamma))$ is defined by
$P_{\mu}(f)(\delta_{t})=\mu(t^{-1}f)\delta_{t},t\in\Gamma$. It is well-known
that $P_{\mu}$ is a $\Gamma$-equivariant unital positive map. Whenever $\mu$
can be contracted using the unitaries, $P_{\mu}$ satisfies a kind of
singularity phenomenon.

######  Lemma 2.6.

SOTconvergence Let $\Gamma$ be a discrete countable group acting on a compact
Hausdorff space $X$. Let $\mu\in\text{Prob}(X)$. Let $\\{u_{n}\\}_{n}$ be a
sequence of unitaries in $L(\Gamma)$ be such that $u_{n}\to 0$ weakly and is
$\mu$-contracting towards $x$. Then,
$u_{n}P_{\mu}(f)u_{n}^{*}\xrightarrow[]{\text{SOT}}f(x)$ for every $f\in
C(X)$.

###### Proof.

Let $\\{u_{n}\\}_{n}\in L(\Gamma)$ be a sequence of unitaries satisfying the
above assumptions. Let us write $u_{n}=\sum_{t\in\Gamma}u_{n}(t)\lambda(t)$,
where $u_{n}(t)=\tau_{0}(u_{n}\lambda(t)^{*})$ for each $t\in\Gamma$.
Moreover, the convergence of the above series is in the $\|\cdot\|_{2}$-norm
induced by the canonical trace $\tau_{0}$. Moreover,

$u_{n}^{*}=\sum_{t\in\Gamma}\overline{u_{n}(t)}\lambda(t^{-1})$

Let $f\in C(X)$ and $\xi\in l^{2}(\Gamma)$ be given. Choose $\epsilon>0$. Let
$M=\max\\{\sup_{t\in\Gamma}|\xi(t)|,1\\}$. Choose $\epsilon^{\prime}$ such
that $2\epsilon^{\prime}M\|f\|_{\infty}<\epsilon/2$. Let $F$ be a finite
subset of $\Gamma$ such that

$\sum_{t\not\in F}|\xi(t)|^{2}<(\epsilon^{\prime})^{2}.$

Choose also $\epsilon^{\prime\prime}>0$ such that

$(\epsilon^{\prime\prime}2\|f\|_{\infty}+\epsilon^{\prime\prime})M|F|<\frac{\epsilon}{2}.$

Let $A=\\{s\in\Gamma:s^{-1}t\mu(f)-f(x)|<\epsilon\leavevmode\nobreak\ \forall
t\in F\\}$. Since $u_{n}$ is $\mu$-contracting towards $x\in X$, we can find a
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$, we have

$\left\|u_{n}|_{A}\right\|_{2}>1-\epsilon^{\prime\prime}.$

We now see that

$\displaystyle\left\|\sum_{t\in\Gamma}\left(u_{n}P_{\mu}(f)u_{n}^{*}\delta_{t}-f(x)\right)\xi(t)\delta_{t}\right\|_{2}$
$\displaystyle\ltx@label{rel1}\overset{(i)}{\leq}\left\|\sum_{t\in
F}\left(u_{n}P_{\mu}(f)u_{n}^{*}\delta_{t}-f(x)\right)\xi(t)\delta_{t}\right\|_{2}+\left\|\sum_{t\not\in
F}\left(u_{n}P_{\mu}(f)u_{n}^{*}\delta_{t}-f(x)\right)\xi(t)\delta_{t}\right\|$

Let us observe that

$\left\|\sum_{t\not\in
F}\left(u_{n}P_{\mu}(f)u_{n}^{*}\delta_{t}-f(x)\right)\xi(t)\delta_{t}\right\|_{2}\ltx@label{rel2}\overset{(ii)}{\leq}2\|f\|_{\infty}\sqrt{\sum_{t\not\in
F}|\xi(t)|^{2}}.$

On the other hand,

$\displaystyle\left\|\sum_{t\in
F}\left(u_{n}P_{\mu}(f)u_{n}^{*}-f(x)\right)\xi(t)\delta_{t}\right\|_{2}$
$\displaystyle\leq\sum_{t\in
F}\left\|\left(u_{n}P_{\mu}(f)u_{n}^{*}-f(x)\right)\xi(t)\delta_{t}\right\|_{2}$
$\displaystyle=\sum_{t\in
F}\|(P_{\mu}(f)u_{n}^{*}-f(x)u_{n}^{*})\xi(t)\delta_{t}\|_{2}$
$\displaystyle\leq\sum_{t\in F}\left(\left\|\sum_{s\notin
A}\overline{u_{n}(s)}(P_{\mu}(f)-f(x))\lambda(s^{-1})\xi(t)\delta_{t}\right\|_{2}+\left\|\sum_{s\in
A}\overline{u_{n}(s)}(P_{\mu}(f)-f(x))\lambda(s^{-1})\xi(t)\delta_{t}\right\|_{2}\right)$
$\displaystyle=\sum_{t\in F}\left(\left\|\sum_{s\notin
A}\overline{u_{n}(s)}(P_{\mu}(f)-f(x))\xi(t)\delta_{s^{-1}t}\right\|_{2}+\left\|\sum_{s\in
A}\overline{u_{n}(s)}(P_{\mu}(f)-f(x))\xi(t)\delta_{s^{-1}t}\right\|_{2}\right)$
$\displaystyle=\sum_{t\in F}\left(\left\|\sum_{s\notin
A}\overline{u_{n}(s)}(s^{-1}t\mu(f)-f(x))\xi(t)\delta_{s^{-1}t}\right\|_{2}+\left\|\sum_{s\in
A}\overline{u_{n}(s)}(s^{-1}t\mu(f)-f(x))\xi(t)\delta_{s^{-1}t}\right\|_{2}\right)$
$\displaystyle\ltx@label{rel3}\overset{(iii)}{\leq}\sum_{t\in
F}\left(2\|f\|_{\infty}\sqrt{\sum_{tu^{-1}\notin
A}\left|\overline{u_{n}(tu^{-1})}\right|^{2}|\xi(t)|^{2}}+\sqrt{\sum_{tu^{-1}\in
A}\left|\overline{u_{n}(tu^{-1})}\right|^{2}\left|(u\mu(f)-f(x))\right|^{2}\left|\xi(t)\right|^{2}}\right)$

Since for all $t\in F$, $|(u\mu(f)-f(x))|<\epsilon$ for every $u$ with
$tu^{-1}\in A$ and $\sum_{s\not\in A}|u_{n}(s)|^{2}<\epsilon^{\prime\prime}$,
the inequality (LABEL:rel3) becomes less than or equal to

$\displaystyle\sum_{t\in
F}\left(2\|f\|_{\infty}M\epsilon^{\prime\prime}+\epsilon\sqrt{\sum_{tu^{-1}\in
A}\left|\overline{u_{n}(tu^{-1})}\right|^{2}\left|\xi(t)\right|^{2}}\right)$
$\displaystyle\leq\sum_{t\in
F}\left(2\|f\|_{\infty}M\epsilon^{\prime\prime}+M\epsilon\sqrt{\sum_{tu^{-1}\in
A}\left|\overline{u_{n}(tu^{-1})}\right|^{2}}\right)$
$\displaystyle\ltx@label{rel4}\overset{(iv)}{\leq}|F|M\left(2\|f\|_{\infty}\epsilon^{\prime\prime}+\epsilon^{\prime\prime}\right).$

Hence for every $n\geq n_{0}$, combining the inequalities (LABEL:rel1),
(LABEL:rel2) and (LABEL:rel4), we obtain that

$\|u_{n}P_{\mu}(f)u_{n}^{*}\xi-f(x)\xi\|_{2}\leq
2\|f\|_{\infty}\sqrt{\sum_{t\not\in
F}|\xi(t)|^{2}}+|F|M(2\|f\|_{\infty}\epsilon^{\prime\prime}+\epsilon^{\prime\prime})<\epsilon.$

The claim follows. ∎

##  3\. Crossed product of tracial von Neumann algebras

In this section, we apply the results of Section 2 in order to study the
position of the relative commutants of certain subalgebras of crossed-product
von Neumann algebras.

######  Theorem 3.1.

tracialvna Let $\Gamma$ be a discrete countable group acting on a compact
Hausdorff space $X$. Let $\mu$ be a probability measure on $X$ and let
$(u_{n})$ be a $\mu$-contracting sequence of unitaries in $L\Gamma$. Let
$(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann algebra, and
$\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a trace-preserving
action. Then
$\\{u_{n}\\}^{\prime}\cap(\mathcal{N}\rtimes\Gamma)\subset\mathcal{N}\rtimes\Gamma_{x}$,
where $x$ is determined by the fact that $(u_{n})$ is $\mu$-contracting
towards $x$.

We shall view
$\mathcal{N}\rtimes\Gamma\subset\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)$.
Moreover, Let $X$ be a $\Gamma$-space. We begin by noting that
$P_{\mu}:C(X)\to\ell^{\infty}(\Gamma)$ is the Poisson tranformation defined by
$P_{\mu}(f)(t)=\mu(t^{-1}f),t\in\Gamma$. This gives us a u.c.p
$\Gamma$-equivariant map from $C(X)$ to $\ell^{\infty}(\Gamma)$. We can view
$\ell^{\infty}(\Gamma)$ as multiplication operators on
$\mathbb{B}(\ell^{2}(\Gamma))$. For $f\in\ell^{\infty}(\Gamma)$, the map
$M(f):\ell^{2}(\Gamma)\to\ell^{2}(\Gamma)$ defined by
$M(f)(\delta_{t})=f(t)\delta_{t}$ is linear and bounded. Therefore, we obtain
a u.c.p map $M\circ P_{\mu}:C(X)\to 1\otimes\mathbb{B}(\ell^{2}(\Gamma)$. We
see that every element of $M\circ P_{\mu}(C(X))$ commutes with $\mathcal{N}$.
We will ignore $M$ for the most part and write $P_{\mu}(f)$ for the ease of
notation. We denote by $\tau$, the canonical trace
$\tilde{\tau}\circ\mathbb{E}$. Note that here
$\mathbb{E}:\mathcal{N}\rtimes\Gamma\to\mathcal{N}$ is the canonical
conditional expectation. Moreover, we consider the $\|.\|_{2}$-norm induced by
$\tau$. We also denote by $\mathbb{E}_{x}$, the canonical conditional
expectation from $\mathcal{N}\rtimes\Gamma$ to $\mathcal{N}\rtimes\Gamma_{x}$.

A crucial ingredient in these arguments is that the state obtained in the
limiting stage satisfies some tracial property. While this is automatic in the
case of amenable tracial von Neumann algebras, we cannot expect it to hold for
us. Nonetheless, before we head on to the proof, we show the existence of an
almost-hypertrace, the last technical bit. To show this, we shall use the
well-known fact that given an inclusion of von Neumann algebras
$\mathcal{M}_{1}\subset\mathcal{M}_{2}$, any normal state $\tau$ on
$\mathcal{M}_{1}$ has an extension to a normal state $\tilde{\tau}$ on
$\mathcal{M}_{2}$ (see for example, [38, Exercise-1, Section 5, Chapter-III].

######  Proposition 3.2.

almosthyperstate Let $\Gamma$ be a discrete countable group acting on a
compact Hausdorff space $X$. Let $\mu$ be a proability measure on $X$ and
$(u_{n})\subset L\Gamma$ a $\mu$-contracting sequence for the action on $X$.
Let $(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann algebra, and
$\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a trace-preserving
action. Then, there exists a state $\psi\in
S\left(\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)\right)$
such that $\psi|_{\mathcal{N}\rtimes\Gamma}=\tau$,
$\psi|_{P_{\mu}(C(X))}=\delta_{x}$, and
$\psi\left(aP_{\mu}(f)b\right)=\psi\left(P_{\mu}(f)ab\right)$ for all
$a\in\\{u_{n}\\}^{\prime}$ and for all
$b\in\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)$.
In particular, $P_{\mu}(C(X))$ falls in the multiplicative domain of $\psi$.

###### Proof.

Let $\tau$ denote the canonical trace $\tilde{\tau}\circ\mathbb{E}$. Let
$P_{\mu}:C(X)\rightarrow\mathbb{B}(l^{2}\Gamma)$ be the Poisson map, which is
ucp $\Gamma$-equivariant. Let $u_{n}\in L(\Gamma)$ be a $\mu$-contracting
sequence towards $x$. We identify an operator $T$ on
$\mathbb{B}(\ell^{2}(\Gamma))$ with $\text{id}\otimes T$, which is an operator
on $\mathbb{B}(H\overline{\otimes}\ell^{2}(\Gamma))$. In this way, we view
$\mathbb{B}(\ell^{2}(\Gamma))$ as a $\Gamma$-invariant subalgebra of
$\mathbb{B}(H\overline{\otimes}\ell^{2}(\Gamma))$. Moreover, if
$T_{n}\in\mathbb{B}(\ell^{2}\Gamma)$ is a uniformly bounded sequence such that
$T_{n}\xrightarrow[\mathbb{B}(\ell^{2}\Gamma)]{\text{SOT}}T$, then,
$\text{id}\otimes
T_{n}\xrightarrow[\mathbb{B}(H\overline{\otimes}\ell^{2}\Gamma)]{\text{SOT}}\text{id}\otimes
T$. Therefore, using SOTconvergence, we see that for every $f\in C(X)$,

$u_{n}(P_{\mu}(f))u_{n}^{*}\xrightarrow[\text{SOT}]{n\to\infty}f(x)\cdot 1.$

In particular, since $(u_{n}(P_{\mu}(f))u_{n}^{*})$ is uniformaly bounded,
this implies that

$u_{n}(P_{\mu}(f))(P_{\mu}(f))^{*}u_{n}^{*}\xrightarrow[\text{SOT}]{n\to\infty}f(x)\overline{f(x)}\cdot
1.$

Consider now the (separable) $C^{*}$-algebra $A$ generated by $P_{\mu}(C(X))$.
Extend the trace $\tau$ to a normal state $\phi$ on
$\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}l^{2}\Gamma)$ and
consider, after passing to a subnet if necessary, a weak∗ limit
$\psi:=\lim_{n}\phi\circ ad(u_{n})$ in
$S(\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}l^{2}\Gamma))$.
We see that $P_{\mu}(f)$ is in the multiplicative domain of $\psi$ for every
$f\in C(X)$.
Claim: $\psi((aP_{\mu}(f)-P_{\mu}(f)a)(aP_{\mu}(f)-P_{\mu}(f)a)^{*})=0$ for
all $a$ which commute with $\\{u_{n}:n\in\mathbb{N}\\}$.
Let us observe that

$\displaystyle\psi((aP_{\mu}(f)-P_{\mu}(f)a)(aP_{\mu}(f)-P_{\mu}(f)a)^{*})$
$\displaystyle=\psi(aP_{\mu}(f)P_{\mu}(f)^{*}a^{*})-\psi(P_{\mu}(f)aP_{\mu}(f)^{*}a^{*})$
$\displaystyle-\psi(aP_{\mu}(f)a^{*}P_{\mu}(f)^{*})+\psi(P_{\mu}(f)aa^{*}P_{\mu}(f)^{*}).$

Now, since $\phi$ is normal and $a$ commutes with
$\\{u_{n}:n\in\mathbb{N}\\}$,

$\displaystyle\psi(aP_{\mu}(f)P_{\mu}(f)^{*}a^{*})$
$\displaystyle=\lim_{n}\phi(u_{n}aP_{\mu}(f)P_{\mu}(f)^{*}a^{*}u_{n}^{*})$
$\displaystyle=\lim_{n}\phi(au_{n}P_{\mu}(f)P_{\mu}(f)^{*}u_{n}^{*}a^{*})$
$\displaystyle=|f(x)|^{2}\phi(aa^{*}).$

On the other hand, since $P_{\mu}(f)$ and $P_{\mu}(f)^{*}$ are in the
multiplicative domain, we have
$\psi(P_{\mu}(f)aa^{*}P_{\mu}(f)^{*})=f(x)\overline{f(x)}\psi(aa^{*})$. Since
$a$ commutes with $\\{u_{n}:n\in\mathbb{N}\\}$, we see that
$\psi(aa^{*})=\lim_{n}\phi(u_{n}aa^{*}u_{n}^{*})=\phi(aa^{*})$. Therefore,

$\psi(P_{\mu}(f)aa^{*}P_{\mu}(f)^{*})=f(x)\overline{f(x)}\psi(aa^{*})=|f(x)|^{2}\phi(aa^{*}).$

Arguing similarly, we see that

$\displaystyle\psi(P_{\mu}(f)aP_{\mu}(f)^{*}a^{*})$
$\displaystyle=f(x)\psi(aP_{\mu}(f)^{*}a^{*})$
$\displaystyle=f(x)\lim_{n}\phi(u_{n}aP_{\mu}(f)^{*}a^{*}u_{n}^{*})$
$\displaystyle=f(x)\lim_{n}\phi(au_{n}P_{\mu}(f)^{*}u_{n}^{*}a^{*})$
$\displaystyle=f(x)\overline{f(x)}\phi(aa^{*})$
$\displaystyle=|f(x)|^{2}\phi(aa^{*}).$

It also follows similarly that

$\psi(aP_{\mu}(f)a^{*}P_{\mu}(f)^{*})=|f(x)|^{2}\phi(aa^{*}).$

Consequently, we see that
$\psi((aP_{\mu}(f)-P_{\mu}(f)a)(aP_{\mu}(f)-P_{\mu}(f)a)^{*})=0$ for every
$a\in\\{u_{n}\\}^{\prime}$. Moreover, for every
$b\in\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}l^{2}\Gamma)$
and $a\in L\Lambda^{\prime}$, we see that

$\displaystyle|\psi(P_{\mu}(f)ab-aP_{\mu}(f)b)|^{2}$
$\displaystyle\leq\psi((P_{\mu}(f)a-aP_{\mu}(f))(P_{\mu}(f)a-aP_{\mu}(f))^{*})\psi(b^{*}b)$
$\displaystyle=0$

Therefore, it follows that $\psi(P_{\mu}(f)ab)=\psi(aP_{\mu}(f)b)$ for all
$a\in\\{u_{n}\\}^{\prime}$ and
$b\in\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}l^{2}\Gamma)$.
∎

Our idea of the proof is motivated by [17, Theorem 1.4].

###### Proof of tracialvna.

From almosthyperstate, we can find a state $\psi\in
S\left(\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)\right)$
such that $\psi|_{\mathcal{N}\rtimes\Gamma}=\tau$,
$\psi|_{P_{\mu}(C(X))}=\delta_{x}$, and
$\psi\left(aP_{\mu}(f)b\right)=\psi\left(P_{\mu}(f)ab\right)$ for all
$a\in\\{u_{n}\\}^{\prime}$ and for all
$b\in\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)$.
Let $\mathcal{M}$ denote the von Neumann algebra
$\\{u_{n}\\}^{\prime}\cap(\mathcal{N}\rtimes\Gamma)$. Let $u\in\mathcal{M}$ be
a unitary element. We shall show that $\|\mathbb{E}_{x}(u)\|_{2}=1$ from
whence it will follow that $u\in\mathcal{N}\rtimes\Gamma_{x}$. Let
$\epsilon>0$. Let
$u_{0}=\sum_{i=1}^{n}a_{i}\lambda(s_{i})\in\mathcal{N}\rtimes\Gamma$ be such
that

(1) $\|u^{*}-u_{0}\|_{2}<\epsilon.$

Let us write $F=\\{s_{1},s_{2},\ldots,s_{n}\\}$. Then, we can rewrite

$u_{0}=\sum_{s\in F\cap\Gamma_{x}}a_{s}\lambda(s)+\sum_{s\in
F\cap\Gamma_{x}^{c}}a_{s}\lambda(s)$

In particular, we see that $sx\neq x$ for all $s\in F\cap\Gamma_{x}^{c}$.
Therefore, we can find $f\in C(X)$ with $0\leq f\leq 1$ such that $f(x)=1$ and
$f(sx)=0$ for all $s\in F\cap\Gamma_{x}^{c}$. From our first claim, we can
find a state $\psi$ on
$\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)$
such that $\psi|_{\mathcal{N}\rtimes\Gamma}=\tau$,
$\psi|_{P_{\mu}(C(X))}=\delta_{x}$, and
$\psi\left(aP_{\mu}(f)b\right)=\psi\left(P_{\mu}(f)ab\right)$ for all
$a\in\\{u_{n}\\}^{\prime}$ and for all
$b\in\mathbb{B}(L^{2}(\mathcal{N},\tilde{\tau})\overline{\otimes}\ell^{2}\Gamma)$.
Below, we write $f$ instead of $P_{\mu}(f)$ for ease of notation. Let us now
observe that

$\displaystyle\left|\psi\left(f(uu_{0}-1)\right)\right|$
$\displaystyle\leq\sqrt{\psi\left((uu_{0}-1)^{*}(uu_{0}-1)\right)}\sqrt{\psi(ff^{*})}$
$\displaystyle=\|uu_{0}-1\|_{2}$
$\left(\psi|_{\mathcal{N}\rtimes\Gamma}=\tau\right)$
$\displaystyle\leq\|u^{*}-u_{0}\|_{2}<\epsilon.$

Therefore,

$\displaystyle\left|\psi\left(ufu_{0}\right)\right|$
$\displaystyle=\left|\psi\left((fuu_{0}\right)\right|$
$\displaystyle=\left|\psi\left(f(uu_{0}-1)\right)+\psi\left(f\right)\right|$
$\displaystyle\geq\left|\psi(f)-|\psi\left(f(uu_{0}-1)\right)|\right|\geq
1-\epsilon.$

To reiterate,

(2) $\left|\psi\left(ufu_{0}\right)\right|\geq 1-\epsilon.$

On the other hand,

$\displaystyle\left|\psi\left((ufu_{0}\right)\right|$
$\displaystyle\leq\left|\psi\left(uf\left(\sum_{s\in
F\cap\Gamma_{x}}a_{s}\lambda(s)\right)\right)\right|+\left|\psi\left(uf\left(\sum_{s\in
F\cap\Gamma_{x}^{c}}a_{s}\lambda(s)\right)\right)\right|$
$\displaystyle\leq\left|\psi\left(uf\mathbb{E}_{x}(u_{0})\right)\right|+\sum_{s\in
F\cap\Gamma_{x}^{c}}\left|\psi\left(ufa_{s}\lambda(s)\right)\right|$

Since $f\in C(X)$, $a_{s}\in\mathcal{N}$ and every element of $C(X)$ commutes
with $\mathcal{N}$ (see the paragraph below tracialvna), we see that

$\displaystyle\sum_{s\in
F\cap\Gamma_{x}^{c}}\left|\psi\left(ufa_{s}\lambda(s)\right)\right|$
$\displaystyle=\sum_{s\in
F\cap\Gamma_{x}^{c}}\left|\psi\left(ua_{s}f\lambda(s)\right)\right|$
$\displaystyle=\sum_{s\in
F\cap\Gamma_{x}^{c}}\left|\psi\left(ua_{s}\lambda(s)s^{-1}f\right)\right|$
$\displaystyle=\sum_{s\in
F\cap\Gamma_{x}^{c}}\left|\psi\left(ua_{s}\lambda(s)\right)f(sx)\right|$
$\displaystyle=0$

It follows therefore that
$\left|\psi\left((ufu_{0}\right)\right|\leq\left|\psi\left(uf\mathbb{E}_{x}(u_{0})\right)\right|$.
Combining this along with equation (1), equation (2) and the Cauchy-Schwartz
inequality, we see that

$\displaystyle 1-\epsilon\leq\left|\psi(ufu_{0})\right|$
$\displaystyle\leq\left|\psi\left(uf\mathbb{E}_{x}(u_{0})\right)\right|$
$\displaystyle\leq\sqrt{\psi(uff^{*}u^{*})}\left\|\mathbb{E}_{x}(u_{0})\right\|_{2}$
$\displaystyle\leq\left\|\mathbb{E}_{x}(u)\right\|_{2}+\epsilon$

As a result, it follows that $\left\|\mathbb{E}_{x}(u)\right\|_{2}\geq
1-2\epsilon$. Since $\epsilon>0$ is arbitrary, it follows that
$u\in\mathcal{N}\rtimes\Gamma_{x}$. ∎

We obtain the following as an immediate result.

######  Corollary 3.3.

relativecommutantforgroupvna Let $\Gamma$ be a discrete countable group acting
on a compact Hausdorff space $X$. Let $(u_{n})$ be a $\mu$-contracting
sequence for the action on $X$ for some probability measure $\mu$ on $X$. Then
$\\{u_{n}\\}^{\prime}\cap L\Gamma\subset L(\Gamma_{x})$, where $x$ is
determined by the fact that $u_{n}$ is $\mu$-contracting towards $x$.

######  Example 3.4.

Let $\Gamma$ be a convergence group. Then, in view of Example 2.5, given a
parabolic element $s\in\Gamma$, for every tracial crossed product
$\mathcal{N}\rtimes\Gamma$ and every diffuse subalgebra $\mathcal{M}$ of
$L(\langle s\rangle)$, the relative commutant of $\mathcal{M}$ in
$\mathcal{N}\rtimes\Gamma$ is injective. If $s\in\Gamma$ is loxodromic, then
$L(\langle s\rangle)^{\prime}\cap(\mathcal{N}\rtimes\Gamma)$ is injective.

##  4\. Relative commutants of subgroups of negatively curved groups

In this section, we examine the relative commutants of subgroups inside groups
that satisfy north pole-south pole-dynamics. We begin with the following
singularity phenomenon, which has been exploited in the past to prove rigidity
results (see, for example, [30, 29, 31, 7, 1] etc).

######  Lemma 4.1.

twodelta Let $X$ be a continuous $\Gamma$-space. Let $\tau\in
S\left(C(X)\rtimes_{r}\Gamma\right)$ such that
$\tau|_{C(X)}=a\delta_{x}+(1-a)\delta_{y}$ for some $x\neq y\in X$. Then,
$\tau(\lambda(s))=0$ for all $s\in\Gamma$ with
$s\\{x,y\\}\cap\\{x,y\\}=\emptyset$.

###### Proof.

Let $s\in\Gamma$ be such that $s\\{x,y\\}\cap\\{x,y\\}=\emptyset$. Using
Uryhson’s lemma, we can find a non-negative continuous function $f\in C(X)$
with $0<f<1$ such that $f|_{\\{x,y\\}}=1$ and $f|_{\\{sx,sy\\}}=0$. Using
Cauchy-Schwartz inequality, we obtain

$\displaystyle\left|\tau(f\lambda(s))\right|^{2}$
$\displaystyle=\left|\tau\left(\sqrt{f}\sqrt{f}\lambda(s)\right)\right|^{2}$
$\displaystyle\leq\tau\left(f\right)\tau\left(\lambda(s^{-1})f\lambda(s)\right)$
$\displaystyle=\tau(f)\tau(s^{-1}.f)$

Since $\tau|_{C(X)}=a\delta_{x}+(1-a)\delta_{y}$, we obtain that

$\tau(s^{-1}f)=as^{-1}.f(x)+(1-a)s^{-1}.f(y)=af(sx)+(1-a)f(sy)=0.$

This shows that $\tau(f\lambda(s))=0$. On the other hand, applying Cauchy-
Schwartz inequality again,

$\displaystyle|\tau\left((1-f)\lambda(s)\right)|^{2}$
$\displaystyle=\left|\tau\left(\sqrt{1-f}\sqrt{1-f}\lambda(s)\right)\right|^{2}$
$\displaystyle\leq\tau\left(1-f\right)\tau\left(\lambda(s^{-1})(1-f)\lambda(s)\right)$
$\displaystyle=\tau(1-f)\tau(s^{-1}.(1-f))$

Let us now see that

$\tau(1-f)=a(1-f(x))+(1-a)(1-f(y))=a(0)+(1-a)(0)=0$

Therefore, we obtain that $\tau\left((1-f)\lambda(s)\right)=0$. Now, combining
the above two identities, we see that

$\tau(\lambda(s))=\tau(f\lambda(s))+\tau\left((1-f)\lambda(s)\right)=0.$

This completes the proof. ∎

An action $\Gamma\curvearrowright X$ is said to have north pole-south pole-
dynamics, if for every infinite order element $g\in\Gamma$, there are unique
fixed point $x_{g}^{+}$ and $x_{g}^{-}$ on the $\Gamma$-space $X$ such that
$g^{n}x\xrightarrow[]{n\to\infty}x_{g}^{+}$ for all $x\neq x_{g}^{-}$. We
denote by $E(g)=\text{Stab}_{\Gamma}(\\{x_{g}^{+},x_{g}^{-}\\})$, the set wise
stabilizer of $\\{x_{g}^{+},x_{g}^{-}\\}$. We denote by $\mathbb{E}_{E(s)}$
the canonical conditional expectation from $C_{r}^{*}(\Gamma)$ onto
$C_{r}^{*}(E(s))$. This also extends to a normal trace-preserving conditional
expectation from $L(\Gamma)$ onto $L(E(s))$.

######  Proposition 4.2.

plclosure Let $\Gamma$ be a discrete group admitting a minimal action
$\Gamma\curvearrowright X$ with the north pole-south pole-dynamics. Let
$s\in\Gamma$ be an infinite order element with the property that
$t\\{x_{s}^{+},x_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}=\emptyset$ for all
$t\not\in E(s)$. Then, given $a\in C_{r}^{*}(\Gamma)$ and $\epsilon>0$, we can
find $\\{s_{1},s_{2},\ldots,s_{m}\\}\subset\langle s\rangle$ such that

$\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(a-\mathbb{E}_{E(s)}(a)\right)\lambda(s_{j})^{*}\right\|<\epsilon.$

Before we head on to the proof, let us briefly ponder our strategy, similar to
that of [37]. Let $\Lambda=\langle s\rangle$. We shall first show that for
every bounded linear functional $\varphi$ on $S(C_{r}^{*}(\Gamma))$, we can
find a bounded linear functional
$\psi\in\overline{\\{s.\omega:s\in\Lambda\\}}^{\text{weak}^{*}}$ such that
$\psi=\psi\circ\mathbb{E}_{\Lambda}$. Here,
$\mathbb{E}_{\Lambda}:C_{r}^{*}(\Gamma)\to C_{r}^{*}(\langle s\rangle)$ is the
canonical conditional expectation. The claim would then follow by a usual
Hahn-Banach separation argument.

###### Proof.

Let $\Gamma$ be a discrete group admitting a minimal action
$\Gamma\curvearrowright X$ with the north pole-south pole dynamics. Since
$\Gamma\curvearrowright X$ is minimal, we can view $C(X)$ as multiplication
operators on $\mathbb{B}(\ell^{2}(\Gamma))$. Given a bounded linear functional
$\varphi$ on $C_{r}^{*}(\Gamma)$, extend it to a bounded linear functional
$\eta$ on $C(X)\rtimes_{r}\Gamma$. We can write
$\eta=c_{1}\omega_{1}-c_{2}\omega_{2}+ic_{3}\omega_{3}-ic_{4}\omega_{4}$,
where $\omega_{i}\in S(C(X)\rtimes_{r}\Gamma)$ and $c_{i}\in\mathbb{C}$ for
each $i=1,2,3,4$. Let $\nu_{i}=\omega_{i}|_{C(X)}$ for each $i=1,2,3,4$. Since
$s$ is an infinite order element, there are unique fixed points $x_{s}^{+}$
and $x_{s}^{-}$ on the $\Gamma$-space $X$ such that
$s^{n}x\xrightarrow[]{n\to\infty}x_{s}^{+}$ for all $x\neq x_{s}^{-}$. Using
the dominated convergence theorem, it follows that
$s^{n}\nu_{i}\xrightarrow[]{\text{weak}^{*}}a_{i}\delta_{x_{s}^{+}}+(1-a_{i})\delta_{x_{s}^{-}}$,
where $a_{i}=\nu_{i}(X\setminus x_{s}^{-})$. By passing to a subnet (four
times) if required, we can assume that
$s^{n}\omega_{i}\to\omega_{i}^{\prime}\in S(C(X)\rtimes_{r}\Gamma)$ for each
$i=1,2,3,4$. Observe that
$\omega_{i}^{\prime}|_{C(X)}=a_{i}\delta_{x_{s}^{+}}+(1-a_{i})\delta_{x_{s}^{-}}$.
Let
$\eta^{\prime}=c_{1}\omega_{1}^{\prime}-c_{2}\omega_{2}^{\prime}+ic_{3}\omega_{3}^{\prime}-ic_{4}\omega_{4}^{\prime}$.
Let $\psi=\eta^{\prime}|_{C_{r}^{*}(\Gamma)}$. We claim that
$\psi=\psi\circ\mathbb{E}_{E(s)}$. Note that
$t\\{x_{s}^{+},x_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}=\emptyset$ for all
$t\not\in E(s)$. It now follows from twodelta that
$\omega_{i}^{\prime}(\lambda(t))=0$ for all $t\not\in E(s)$ and for all
$i=1,2,3,4$. This shows that
$\omega_{i}^{\prime}|_{C_{r}^{*}(\Gamma)}=\omega_{i}^{\prime}\circ\mathbb{E}_{E(s)}$
for each $i=1,2,3,4$. Consequently, it follows that
$\psi=\psi\circ\mathbb{E}_{E(s)}$. The claim now follows by a usual Hahn-
Banach separation argument (see, for example, [21, Theorem 3.4]). ∎

It was shown in [2] that the averaging scheme at the level of the group
$C^{*}$-algebra lifts to the same averaging scheme at the level of the crossed
product. We merely reiterate the steps to prove that the averaging established
in plclosure lifts to the crossed product of tracial von Neumann algebras.
Given a tracial von Neumann algebra $(\mathcal{N},\tilde{\tau})$ and a trace
preserving action $\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$, we let
$\tau=\tilde{\tau}\circ\mathbb{E}$ which is a faithful normal trace on
$\mathcal{N}\rtimes\Gamma$. We do all the approximations in the
$\|\cdot\|_{2}$-norm induced by $\tau$. We denote by $\mathbb{E}$, the
canonical conditional expectation from $\mathcal{N}\rtimes\Gamma$ onto
$\mathcal{N}$. Moreover, we shall use $\tilde{\mathbb{E}}_{E(s)}$ to denote
the canonical conditional expectation from $\mathcal{N}\rtimes\Gamma$ onto
$\mathcal{N}\rtimes E(s)$.

######  Theorem 4.3.

avgtracialvna Let $\Gamma$ be a discrete group admitting a minimal action
$\Gamma\curvearrowright X$ with the north pole-south pole-dynamics. Let
$s\in\Gamma$ be an infinite order element with the property that
$t\\{x_{s}^{+},x_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}=\emptyset$ for all
$t\not\in E(s)$. Let $(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann
algebra and $\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a trace-
preserving action. Let $\mathcal{M}=\mathcal{N}\rtimes\Gamma$. Then, given
$x\in\mathcal{M}$ and $\epsilon>0$, we can find
$\\{s_{1},s_{2},\ldots,s_{m}\\}\subset\langle s\rangle$ such that

$\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(x-\tilde{\mathbb{E}}_{E(s)}(x)\right)\lambda(s_{j})^{*}\right\|_{2}<\epsilon.$

###### Proof.

Let $x\in\mathcal{M}$ and $\epsilon>0$ be given. We can find a finite set
$F\subset\Gamma$ and finitely many elements $\\{a_{t}:t\in
F\\}\subset\mathcal{N}$ such that

$\left\|x-\sum_{t\in F}a_{t}\lambda(t)\right\|_{2}<\frac{\epsilon}{3}.$

Since $\mathbb{E}\circ\tilde{\mathbb{E}}_{E(s)}=\mathbb{E}$, it follows that

$\left\|\tilde{\mathbb{E}}_{E(s)}(x)-\sum_{t\in E(s)\cap
F}a_{t}\lambda(t)\right\|_{2}<\frac{\epsilon}{3}.$

Let $M=\sup_{t\in F}\|a_{t}\|$. Using plclosure for $a=\sum_{t\in E(s)^{c}\cap
F}\lambda(t)$, we can find $\\{s_{1},s_{2},\ldots,s_{m}\\}\subset\langle
s\rangle$ such that

$\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(\sum_{t\in E(s)^{c}\cap
F}\lambda(t)\right)\lambda(s_{j})^{*}\right\|<\frac{\epsilon}{3|F|M}.$

It follows from [37, Lemma 4.1] that

$\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\lambda(t)\lambda(s_{j})^{*}\right\|_{2}\leq\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\lambda(t)\lambda(s_{j})^{*}\right\|<\frac{\epsilon}{3|F|M},\leavevmode\nobreak\
\forall t\in E(s)^{c}\cap F.$

Therefore, using [2, Lemma 2.1], we obtain that

$\displaystyle\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(\sum_{t\in
E(s)^{c}\cap F}a_{t}\lambda(t)\right)\lambda(s_{j})^{*}\right\|$
$\displaystyle\leq\sum_{t\in E(s)^{c}\cap
F}\|a_{t}\|\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\lambda(t)\lambda(s_{j})^{*}\right\|$
$\displaystyle\leq\sum_{t\in E(s)^{c}\cap
F}\|a_{t}\|\frac{\epsilon}{3|F|M}<\frac{\epsilon}{3}.$

Putting all these together, along with an application of triangle inequality,
we see that

$\displaystyle\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(x-\tilde{\mathbb{E}}_{E(s)}(x)\right)\lambda(s_{j})^{*}\right\|_{2}$
$\displaystyle\leq\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(x-\sum_{t\in
F}a_{t}\lambda(t)\right)\lambda(s_{j})^{*}\right\|_{2}$
$\displaystyle+\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(\sum_{t\in
F\cap
E(s)}a_{t}\lambda(t)-\tilde{\mathbb{E}}_{E(s)}(x)\right)\lambda(s_{j})^{*}\right\|_{2}$
$\displaystyle+\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(\sum_{t\in
E(s)^{c}\cap F}a_{t}\lambda(t)\right)\lambda(s_{j})^{*}\right\|_{2}$
$\displaystyle\leq\left\|x-\sum_{t\in
F}a_{t}\lambda(t)\right\|_{2}+\left\|\sum_{t\in F\cap
E(s)}a_{t}\lambda(t)-\tilde{\mathbb{E}}_{E(s)}(x)\right\|_{2}+\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(\sum_{t\in
E(s)^{c}\cap F}a_{t}\lambda(t)\right)\lambda(s_{j})^{*}\right\|$
$\displaystyle\leq\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon.$

The claim follows. ∎

Consequently, we can determine the position of the relative commutants of
specific subgroups of a non-elementary acylindrically-hyperbolic group with at
least one infinite order element. We briefly recall the definitions and refer
the readers to [34] for more details.

### Acylindrically Hyperbolic groups

An action $\Gamma\curvearrowright(X,d)$ on a metrizable space is considered
acylindrical if for every $\epsilon>0$, there exist $\delta,N>0$ such that for
any $x,y\in X$ with $d(x,y)\geq\delta$, the number of elements $g\in\Gamma$
satisfying $d(x,gx)\leq\epsilon$ and $d(y,gy)\leq\epsilon$ is at most $N$. A
group $\Gamma$ is called acylindrically hyperbolic if it admits a non-
elementary acylindrical action on a hyperbolic space.

Every non-elementary hyperbolic group is acylindrically hyperbolic. Further
examples of acylindrically hyperbolic groups include non-(virtually) cyclic
groups hyperbolic relative to proper subgroups, $\text{Out}(F_{n})$ for $n>1$,
many mapping class groups, and non-(virtually cyclic) groups acting properly
on proper CAT($0$)-spaces and containing rank one elements, among others (for
more details, refer to [34, Section 8] and the references therein).

For a group $\Gamma$ acting on a hyperbolic space $S$, an infinite order
element $g\in\Gamma$ is called loxodromic if it has precisely two fixed points
$x_{g}^{+},x_{g}^{-}$ on the Gromov boundary $\partial S$ and $g^{n}x\to
x_{g}^{+}$ for every $x\in\partial S$ except $x_{g}^{-}$. It turns out that a
group being acylindrically hyperbolic is equivalent to the notion of weak
proper discontinuity introduced by Bestvina and Fujiwara [16]. Let $\Gamma$ be
a group acting on a hyperbolic space $S$. An element $g\in\Gamma$ is said to
have the weak proper discontinuity property (in this case, we say that $g$ is
a WPD element) if for every $\epsilon>0$ and every $x\in S$, there exist
$M\in\mathbb{N}$ such that the number of elements $h\in\Gamma$ satisfying
$d(x,hx)<\epsilon$ and $d(g^{M}x,hg^{M}x)<\epsilon$ is finite.

Osin [34, Theorem 1.2] later established that a group $\Gamma$ being
acylindrically hyperbolic is equivalent to the existence of a loxodromic
element $g\in\Gamma$ that satisfies the weak proper discontinuity condition.
Moreover, there is a unique maximal virtually cyclic subgroup $E(g)\leq\Gamma$
containing $g$. Explicitly,
$E(g)=\text{Stab}_{\Gamma}(\\{x_{g}^{+},x_{g}^{-}\\})$ is the set wise
stablizer of $\\{x_{g}^{+},x_{g}^{-}\\}$ (see for example, [24, Lemma 6.5]).

######  Corollary 4.4.

relcommutantinhyp Let $\Gamma$ be a non-elementary acylindrically hyperbolic
group. Let $\Lambda\leq\Gamma$ be a subgroup with one infinite order WPD
loxodromic element $s\in\Lambda$. Assume that
$\\{tx_{s}^{+},tx_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}=\emptyset$ for all
$t\not\in E(s)$. Let $(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann
algebra and $\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a trace-
preserving action. Let $\mathcal{M}=\mathcal{N}\rtimes\Gamma$. Then,
$L(\Lambda)^{\prime}\cap\mathcal{M}\subset\mathcal{N}\rtimes E(s)$.

###### Proof.

Since $s$ is an infinite order WPD loxodromic element, it satisfies the north
pole-south pole dynamics on the Gromov boundary. Let $x\in
L(\Lambda)^{\prime}\cap\mathcal{M}$. Let $s\in\Lambda$ be an infinite order
loxodromic element. Let $\epsilon>0$. Using avgtracialvna, it follows that we
can find $s_{1},s_{2},\ldots,s_{m}\subset\langle s\rangle$ such that

(3)
$\left\|\frac{1}{m}\sum_{j=1}^{m}\lambda(s_{j})\left(x-\tilde{\mathbb{E}}_{E(s)}(x)\right)\lambda(s_{j})^{*}\right\|_{2}<\epsilon.$

Note that here
$\tilde{\mathbb{E}}_{E(s)}:\mathcal{N}\rtimes\Gamma\to\mathcal{N}\rtimes E(s)$
is the canonical conditional expectation. Let us write
$\tilde{\mathbb{E}}_{E(s)}(x)=\sum_{t\in E(s)}a_{t}\lambda(t)$, where the
convergence is in the $\|\cdot\|_{2}$-norm. For any $s_{j}\in\langle
s\rangle$, writing it as $s^{m_{j}}$ for some $m_{j}\in\mathbb{Z}$, we see
that

$\lambda(s_{j})\tilde{\mathbb{E}}_{E(s)}(x)\lambda(s_{j})^{*}=\sum_{t\in
E(s)}\alpha_{s_{j}}(a_{t})\lambda(s_{j}ts_{j}^{-1})=\sum_{t\in
E(s)}\alpha_{s_{j}}(a_{t})\lambda(s^{m_{j}}ts^{-m_{j}}).$

Since $t\in E(s)=\\{x_{s}^{+},x_{s}^{-}\\}$, we see that
$s^{m_{j}}ts^{-m_{j}}\\{x_{s}^{+},x_{s}^{-}\\}=\\{x_{s}^{+},x_{s}^{-}\\}$. As
such, we can now see that
$\lambda(s_{j})\tilde{\mathbb{E}}_{E(s)}(x)\lambda(s_{j})^{*}\in\mathcal{N}\rtimes
E(s)$ for each $j=1,2,\ldots,m$. Writing

$\sum_{j=1}^{m}\lambda(s_{j})\tilde{\mathbb{E}}_{E(s)}(x)\lambda(s_{j})^{*}=y_{E(s)},$

since $x\in L(\Lambda)^{\prime}\cap\mathcal{M}$, it follows from equation (3)
that

$\left\|x-y_{E(s)}\right\|_{2}<\epsilon.$

Since $\epsilon>0$ is arbitrary, it is evident that $x\in\mathcal{N}\rtimes
E(s)$. Therefore, $L(\Lambda)^{\prime}\cap\mathcal{M}\subset\mathcal{N}\rtimes
E(s)$. ∎

In addition, if we assume that $\mathcal{N}$ is amenable, then it can be
concluded that the relative commutant $L(\Lambda)^{\prime}\cap\mathcal{M}$ is
amenable.

######  Example 4.5.

Let $\Gamma$ be a torsion-free hyperbolic group and $\Lambda\leq\Gamma$ be an
infinite subgroup. Then, $\Lambda$ contains an element of infinite order and
is loxodromic. Let’s call it $s$. Denote by $x_{s}^{+}$ and $x_{s}^{-}$, the
corresponding fixed points on the Gromov boundary $\partial\Gamma$. It is
well-known that
$E(s)=\text{Stab}_{\Gamma}\left(\\{x_{s}^{+},x_{s}^{-}\\}\right)$ (see for
example [24, Lemma 6.5]). Since $s\in\text{Fix}(x_{s}^{+})$ is a hyperbolic
element, and $x_{s}^{-}$ is the unique fixed point of $x$ in
$\partial\Gamma\setminus x_{s}^{+}$, it follows from the proof of [25, Theorem
8.30] that $\text{Fix}(x_{s}^{+})=\text{Fix}(x_{s}^{-})$. We claim that
$\\{tx_{s}^{+},tx_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}=\emptyset$ for all
$t\not\in E(s)$. Let $t\not\in E(s)$. Since
$\text{Fix}(x_{s}^{+})=\text{Fix}(x_{s}^{-})$, it follows that $tx_{s}^{+}\neq
x_{s}^{+}$ and $tx_{s}^{-}\neq x_{s}^{-}$. If $tx_{s}^{+}=x_{s}^{-}$, then
$t^{-1}stx_{s}^{+}=x_{s}^{+}$. Therefore, $t^{-1}st\in\text{Fix}(x_{s}^{-})$.
Therefore, we see that $t^{-1}stx_{s}^{-}=x_{s}^{-}$. This further implies
that $s(tx_{s}^{-})=(tx_{s}^{-})$. Since $s$ is a loxodromic element, either
$tx_{s}^{-}=x_{s}^{-}$ or $tx_{s}^{-}=x_{s}^{+}$. If $tx_{s}^{-}=x_{s}^{-}$,
it would follow that $tx_{s}^{+}=x_{s}^{-}=tx_{s}^{-}$ from whence we would
obtain that $x_{s}^{+}=x_{s}^{-}$ which would contradict the fact that $s$ is
an infinite loxodromic element. Therefore, $tx_{s}^{-}=x_{s}^{+}$. This shows
that $t\in E(s)$ which contradicts our earlier choice of $t\not\in E(s)$. If
$tx_{s}^{-}=x_{s}^{+}$, the argument follows analogously by replacing
$t^{-1}st$ with $tst^{-1}$. As such, we can now apply relcommutantinhyp to
conclude the relative commutant
$L(\Lambda)^{\prime}\cap\mathcal{N}\rtimes\Gamma$ is contained inside
$\mathcal{N}\rtimes E(s)$ for any trace-preserving action
$\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$. Under the further
assumption of amenability of $\mathcal{N}$, it follows that
$L(\Lambda)^{\prime}\cap\mathcal{N}\rtimes\Gamma$ is amenable since $E(s)$ is
an amenable subgroup of $\Gamma$.

There are many acylindrically hyperbolic groups for which we can find an
element $t\not\in E(s)$ such that
$t\\{x_{s}^{+},x_{s}^{-}\\}\cap\\{x_{s}^{+},x_{s}^{-}\\}\neq\emptyset$.
Nevertheless, we can still determine the position of the relative commutant of
any diffuse von Neumann subalgebra of $L(\langle s\rangle)$ in these
situations. Recall that for an action $\Gamma\curvearrowright X$ with north
pole-south pole-dynamics, an element $s\in\Gamma$ is called parabolic if there
exists a unique fixed point $x_{s}^{+}\in X$ such that both
$\\{s^{n}x\\}_{n\in\mathbb{N}}$ and $\\{s^{-n}x\\}_{n\in\mathbb{N}}$ converge
to $x_{s}^{+}$ as $n\to\infty$ for every $x\in X$.

######  Corollary 4.6.

diffnorthsouth Let $\Gamma$ be a discrete group admitting an action
$\Gamma\curvearrowright X$ with the north pole-south pole-dynamics. Let
$s\in\Gamma$ be an infinite order parabolic element. Suppose there exists a
quasi-invariant probability measure $\mu\in\text{Prob}(X)$ such that
$\mu(x_{s}^{+})\neq 0$. Let $(\mathcal{N},\tilde{\tau})$ be a tracial von
Neumann algebra and $\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$ be a
trace-preserving action. Let $\mathcal{M}=\mathcal{N}\rtimes\Gamma$. Then,
$\mathcal{M}_{1}^{\prime}\cap\mathcal{M}\leq\mathcal{N}\rtimes\Gamma_{x_{s}^{+}}$
for any diffuse subalgebra $\mathcal{M}_{1}\leq L(\langle s\rangle)$.

###### Proof.

Let $s\in\Gamma$ be an infinite order parabolic element. By assumption, there
exists one fixed point $x_{s}^{+}$ on $X$. Moreover,
$s^{n}y\xrightarrow{n\to\infty}x_{s}^{+}$ for all $y\in X$. Let
$P_{\mu}:C(X)\to\mathbb{B}(\ell^{2}(\Gamma))$ be the associated Poisson-
transformation. It follows from the definition 2.1 that $(s^{n})$ is $\mu$
contracting towards $x_{s}^{+}$. In particular, every diverging sequence
$(\lambda_{n})\subset\Lambda=\langle s\rangle$ is $\mu$ contracting towards
$x_{s}^{+}$. Let $\mathcal{M}_{1}\leq L(\langle s\rangle)$ be a diffuse
subalgebra. Let $(u_{n})\subset\mathcal{U}(\mathcal{M}_{1})$ be a sequence of
unitaries which converge to $0$ weakly. We can now appeal to Lemma 2.4 to
conclude that $u_{n}$ is $\mu$-contracting towards $x_{s}^{+}$. The claim now
follows from tracialvna. ∎

Recall that a CAT$(0)$-cube complex is a simply connected cell complex whose
cells are Euclidean cubes $[0,1]^{d}$ of various dimensions. We refer the
readers to [22], [19], and [23] for more details on these. One can assign many
compact Hausdorff boundaries to a CAT$(0)$-cube complex (see, for example [33,
Section 1.3]). For our purposes, given a CAT$(0)$ metric space, we consider
the action on the visual boundary $\partial X$ (see [19, Chapter 8]) equipped
with the cone-topology. If $X$ is Gromov hyperbolic, then $\partial X$ is the
classical Gromov boundary of $X$.

Let $\Gamma$ be a countable discrete group acting on a proper CAT$(0)$ cube
complex $X$ (not necessarily hyperbolic) by isometries. We say that the action
is elementary if the limit set $LX$ (the set of accumulation points in
$\partial X$ of an orbit of the action) consists of at most two points or if
$\Gamma$ fixes a point on $\partial X$.

######  Example 4.7 (CAT$(0)$-cube complexes).

Let $\Gamma$ be a countable discrete group acting on a proper CAT$(0)$ cube
complex $X$ (not necessarily hyperbolic) by isometries in a non-elementary
way. Let $s\in\Gamma$ be rank-one isometry. It is well-known that any rank-one
isometry $g\in\text{Isom}(X)$ has the north pole-south pole-dynamics (see, for
example, [28, Lemma 4.4]). Using [28, Theorem 1.1], we see that the limit set
$LX\subset\partial X$ is perfect. It follows from [18, Lemma 2.1] that there
is a non-atomic measure $\mu\in\text{Prob}(\partial X)$. Since $s$ is a rank-
one isometry, there are two fixed points $x_{s}^{+}$ and $x_{s}^{-}$ on the
visual boundary $\partial X$. Moreover,
$s^{n}y\xrightarrow{n\to\infty}x_{s}^{+}$ for all $y\neq x_{s}^{-}\in\partial
X$. Since $\mu$ is non-atomic, we see that $\mu(x_{s}^{-})=0$. Now, let
$(\mathcal{N},\tilde{\tau})$ be a tracial von Neumann algebra, and
$\Gamma\curvearrowright(\mathcal{N},\tilde{\tau})$, a trace-preserving action.
Setting $\mathcal{M}=\mathcal{N}\rtimes\Gamma$, it follows from tracialvna
that $L\left(\langle
s\rangle\right)^{\prime}\cap\mathcal{M}\leq\mathcal{N}\rtimes\Gamma_{x_{s}^{+}}$.
If we further assume that $\mathcal{N}$ is amenable, then in this case, since
$\Gamma_{x_{s}^{+}}$ is amenable (see the argument in the last paragraph of
[32, Lemma 5.6]), we conclude that $L\left(\langle
s\rangle\right)^{\prime}\cap\mathcal{M}$ is amenable.

It is not difficult to find actions $\Gamma\curvearrowright X$ on
CAT$(0)$-cube complexes $X$ which are non-elementary. For example, if
$|\partial X|>2$ and $\Gamma\curvearrowright X$ cocompactly by isometries,
then the action is necessarily non-elementary (see for example [8]).

##  5\. The case of $\operatorname{SL}(d,\mathbb{\mathbb{Z}})$

This section applies our results to the von Neumann algebras associated with
infinite subgroups of $\operatorname{SL}(d,\mathbb{Z})$, $d\geq 2$. We show
that for each such subgroup $\Lambda$, the relative commutant of $L\Lambda$ is
always contained in the von Neumann algebra of the intersection of some
parabolic subgroup with $\operatorname{SL}(d,\mathbb{Z})$. In the case $d=3$
or if the subgroup is Zariski dense in $\operatorname{SL}(d,\mathbb{R})$, such
parabolic subgroups are always Borel groups.

######  Proposition 5.1.

Let $d\in\mathbb{N}$ and $\Gamma$ be an infinite subgroup of
$\operatorname{SL}(d,\mathbb{Z})$. Then there is a parabolic subgroup $P$ of
$\operatorname{SL}(d,\mathbb{R})$ such that $L\Gamma^{\prime}\cap
L\operatorname{SL}(d,\mathbb{Z})\subset L(\operatorname{SL}(d,\mathbb{Z})\cap
P)$.

###### Proof.

Let $G=\operatorname{SL}(d,\mathbb{R})$. We want to show that given a
diverging sequence $(\gamma_{n})$ in $\operatorname{SL}(d,\mathbb{Z})$ there
are a parabolic subgroup $P$ of $G$, an
$\operatorname{SL}(d,\mathbb{Z})$-quasi-invariant probability measure on $G/P$
and a point $y\in G/P$ such that, up to taking a subsequence, for $\mu$-almost
every point $x$ in $G/P$ we have $\lim_{n}\gamma_{n}x=y$.
Let then $(\gamma_{n})$ be such a sequence and write
$\gamma_{n}=k_{n}a_{n}k_{n}^{\prime}$ ($KAK$ decomposition in
$\operatorname{SL}(d,\mathbb{R})$), in such a way that the diagonal entries
$(\lambda_{i}^{(n)})$ of $a_{n}$ are taken in decreasing order:
$\lambda_{i}(n)\geq\lambda_{i+1}^{(n)}$ for every $n$, for every $i=1,...,d$.
Up to taking a subsequence we can suppose that
$\lambda_{i}^{(n)}/\lambda_{i+1}^{(n)}$ converges to a point in $(0,\infty]$
for every $i=1,...,d-1$, and $k_{n}\rightarrow k$, $k_{n}^{\prime}\rightarrow
k^{\prime}$ in $K$. We consider the partition of $\\{1,...,d\\}$ into
$I_{1},...,I_{l}$ subsets (for some $l\in\mathbb{N}$) defined by the condition
that $i$ and $i+j$ belong to the same set $I_{m}$ if and only if
$\lambda_{i}^{(n)}/\lambda_{i+j}^{(n)}$ converges to a finite number. Then we
consider the parabolic subgroup $P$ associated with this partition, i.e., the
one given by matrices in $\operatorname{SL}(d,\mathbb{R})$ of the form

$\left(\begin{array}[]{ccccc}GL_{|I_{1}|}(\mathbb{R})&*&*&...&*\\\
0&GL_{|I_{2}|}(\mathbb{R})&*&&\vdots\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&...&0&GL_{|I_{l}|}(\mathbb{R})\end{array}\right).$

Let now $A=\\{g\in\operatorname{SL}(d,\mathbb{R})\;|\;\det(g_{i})\neq
0\;\forall i=1,...,d-1\\}$, where $g_{i}$ is the $i$-th principal minor of
$g$. $A$ is a dense open subset of $G$. It follows from Gaussian elimination
that every element of $A$ can be written as a product of an element of the
group $T$ of strictly lower triangular matrices (i.e., the ones having only
$1$’s on the diagonal) and an element from the Borel subgroup $B$ of upper
triangular matrices (cf. the proof of [40] Lemma 5.1.4). The map $T\rightarrow
A/P$ is continuous and surjective; it restricts to a continuous surjective map
$T\backslash\\{T\cap P\\}\rightarrow A/P\backslash\\{eP\\}$. Let then $x\in
T\backslash\\{T\cap P\\}$ and write it as $x=(X_{i,j})_{i,j=1}^{l}$, where
$X_{i,j}$ is a matrix of size $|I_{i}|\times|I_{j}|$; in the same way we write
$a_{n}=(\Lambda_{i,j})_{i,j=1}^{l}$. Define the sequence in $P$ given by
$h_{n}=(H_{i,j})_{i,j=1}^{l}$, where
$H_{i,j}=\delta_{i,j}(\Lambda_{i,i}X_{i,i})^{-1}$. Then
$a_{n}xh_{n}\rightarrow e$ and so $a_{n}xP\rightarrow eP$. Let now $yP\in A/P$
and $C\subset A/P$ be a compact neighborhood of $yP$. Let $U$ be an open
subset around $eP$ with an empty intersection with $C$. For every $xP\in C$
there is $n_{xP}$ such that $a_{n_{x}P}xP\in U$. Hence the open sets
$a_{n_{xP}}^{-1}U$ cover $C$. It follows that the sequence
$k^{-1}k_{n}a_{n}k_{n}^{\prime}(k^{\prime})^{-1}xP$ converges to $eP$. Hence,
the result follows by choosing any $\operatorname{SL}(d,\mathbb{Z})$-quasi-
invariant probability measure on $G/P$ which gives zero mass to $G/P\backslash
A/P$. ∎

######  Proposition 5.2.

Let $n\geq 2$ and let $\Gamma=SL(n,\mathbb{Z})$. Let also $\mu$ be the
$K$-invariant probability measure on the complete $n$-dimensional flag
variety. Then, every subgroup of $\Gamma$, which is Zariski dense in
$SL(n,\mathbb{R})$, contains a $\mu$-contracting sequence.

###### Proof.

If $\Lambda$ is a Zariski dense subgroup of $\Gamma$, then we can apply the
procedure in [26, Theorem 3.6] (since the action of
$\operatorname{SL}(n,\mathbb{Z})$ on the complete flag variety is transitive)
to deduce that $\Lambda$ has the contraction property (as defined in [26,
Definition 3.1]. The result follows from [26, Lemma 3.9] and the proof of
Proposition 5.1.∎

######  Corollary 5.3.

Let $\Lambda\subset\operatorname{SL}(d,\mathbb{\mathbb{Z}})$ be a Zariski-
dense subgroup of $\operatorname{SL}(d,\mathbb{R})$. Then
$L\Lambda^{\prime}\cap L\operatorname{SL}(d,\mathbb{Z})\subset L\Gamma_{x}$
for some $x\in\operatorname{SL}(d,\mathbb{R})/B_{d}$ (which is injective),
where $B_{d}$ is the Borel subgroup of upper-triangular matrices in
$\operatorname{SL}(d,\mathbb{R})$. In particular, this applies to infinite
commensurated subgroups of $\operatorname{SL}(d,\mathbb{Z})$.

###### Proof.

Follows from Theorem LABEL:tracialvna and [6, Lemma 7.5]. ∎

A stronger result holds if we assume $d=3$ in the above Proposition.

######  Proposition 5.4.

Let $\Gamma$ be an infinite subgroup of $\operatorname{SL}(3,\mathbb{Z})$.
Then $L\Gamma^{\prime}\cap L\operatorname{SL}(3,\mathbb{Z})\subset
LB\cap\operatorname{SL}(3,\mathbb{Z})$ for some Borel subgroup
$B\subset\operatorname{SL}(3,\mathbb{R})$. In particular, the relative
commutant of every infinite subgroup of $\operatorname{SL}(3,\mathbb{Z})$ is
injective.

###### Proof.

It follows from the discussion in [35, Example 7] that every infinite subgroup
of $\operatorname{SL}(3,\mathbb{Z})$ contains an element whose singular values
are pairwise distinct. The result follows arguing as in the proof of
Proposition 5.1. ∎

######  Example 5.5.

([18] Corollary 6.4) Let $\Lambda$ be an infinite subgroup of $P$, where $P$
is the parabolic subgroup associated to the partition
$\\{\\{1,2\\},\\{3\\}\\}$, such that for every element $g\in\Lambda$, $1$ is a
singular value of $g$. Then the relative commutant of every diffuse subalgebra
of $L\Lambda$ inside $L\operatorname{SL}(3,\mathbb{Z})$ is injective. Indeed,
by the proof of Proposition 5.1, every divergent sequence in $\Lambda$ is
$\mu$-contracting towards $eP$. It follows from Lemma 2.4 and Theorem
LABEL:tracialvna that the relative commutant of every diffuse von Neumann
subalgebra of $L\Lambda$ is contained in $LP$, hence it coincides with its
relative commutant inside $LP$. But $LP$ is solid, and the result follows.
Note that if $\Lambda$ is contained in $\mathbb{Z}^{2}$ (after identifying $P$
with $\operatorname{SL}(2,\mathbb{Z})\rtimes\mathbb{Z}^{2}$), the result
follows from another application of Theorem LABEL:tracialvna.

We can give a more general example in the case when $d\geq 2$.

######  Example 5.6.

If $d\geq 2$ and $\Lambda$ is an infinite subgroup of
$\operatorname{SL}(d,\mathbb{Z})$ with the property that for every
$g\in\Lambda$, $g$ has only two singular values which are not $1$, then the
relative commutant of every diffuse subalgebra of $L\Lambda$ is contained in
the von Neumann algebra of the parabolic subgroup associated to the partition
$\\{\\{1,...,d-1\\},\\{d\\}\\}$. This is for example the case for certain
embeddings of $\operatorname{SL}(2,\mathbb{Z})$ inside
$\operatorname{SL}(d,\mathbb{Z})$ for $d\geq 2$.

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# Charged Lepton-Flavor Violation in Beyond-Standard Models

Junji Hisano  Institute for Cosmic Ray Research (ICRR), University of Tokyo,
Kashiwa, Chiba 277-8582, Japan

###### Abstract

I discuss charged lepton-flavor violation in physics beyond the standard model
and review topics related to it.

##  I Introduction

It is found from discovery of the neutrino oscillation that the lepton-flavor
symmetries are not exact in nature, while the charged lepton-flavor violation
(cLFV) has not been observed yet. The cLFV is suppressed even if the tiny
neutrino masses observed in the experiments are introduced in the standard
model (SM). The cLFV processes are proportional to the fourth power of the
neutrino masses due to the GIM mechanism, similar to the quark sector. The
predicted branching ratio of $\mu\rightarrow e\gamma$ is smaller than
$10^{-54}$. However, the lepton-flavor symmetries are accidental in the SM,
and it is a big mystery in particle physics why the cLFV processes are not
discovered, when considering beyond-standard models (BSMs) Raidal:2008jk .

Searches for cLFV processes have long history. Search for $\mu\rightarrow
e\gamma$ was performed soon after muons were discovered, and it was found that
muons are not an excited state of electrons. On 60’s, the two-neutrino
hypothesis, in which the lepton flavor conservations were introduced, was
proposed to suppress $\mu\rightarrow e\gamma$. The tau-lepton flavor
conservation has been also tested after tau lepton was discovered on 75’.
However, on 98’, the neutrino oscillation was discovered at SuperKamiokande
experiments.

Now the bounds on the cLFV processes are significantly improved by efforts of
experimentalists. The current experimental bounds on the representative cLFV
processes of muon are given as follows,

$\displaystyle{\rm Br}(\mu\rightarrow e\gamma)$ $\displaystyle<$
$\displaystyle 1.2\times 10^{-11}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Brooks:1999pu}{\@@citephrase{(}}{\@@citephrase{)}}}},$ (1)
$\displaystyle{\rm Br}(\mu\rightarrow eee)$ $\displaystyle<$ $\displaystyle
1.0\times 10^{-12}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bellgardt:1987du}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(2) $\displaystyle{\rm R}(\mu\rightarrow e;{\rm Ti})$ $\displaystyle<$
$\displaystyle 6.1\times 10^{-13}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Dohmen:1993mp}{\@@citephrase{(}}{\@@citephrase{)}}}},$ (3)
$\displaystyle{\rm R}(\mu\rightarrow e;{\rm Au})$ $\displaystyle<$
$\displaystyle 7.0\times 10^{-13}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bertl:2006up}{\@@citephrase{(}}{\@@citephrase{)}}}}.$ (4)

$\mu$-$e$ conversion rates in nuclei, given in Eqs. 3 and 4, are normalized by
the muon capture rates. Those for tau-lepton’s processes come from the Belle
and Babar experiments, and they are111 The bounds on Eqs. 5 and 6 are combined
results of Belle Abe:2006sf  and Babar Aubert:2005ye ; Aubert:2005wa .

$\displaystyle{\rm Br}(\tau\rightarrow\mu\gamma)$ $\displaystyle<$
$\displaystyle 1.6\times 10^{-8}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Banerjee:2007rj}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(5) $\displaystyle{\rm Br}(\tau\rightarrow e\gamma)$ $\displaystyle<$
$\displaystyle 9.4\times 10^{-8}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Banerjee:2007rj}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(6) $\displaystyle{\rm Br}(\tau\rightarrow 3l)$ $\displaystyle<$
$\displaystyle\sim 10^{-8}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Banerjee:2007rj}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(7) $\displaystyle{\rm Br}(\tau\rightarrow l+{\rm hadron(s)})$
$\displaystyle<$ $\displaystyle\sim
10^{-8}~{}\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Banerjee:2007rj}{\@@citephrase{(}}{\@@citephrase{)}}}}.$
(8)

A new experiment for $\mu\rightarrow e\gamma$ search, MEG, has been started at
PSI this year meg . It is argued tmori  that a successful physics run for 5-6
months reaches to $(7$-$8)\times 10^{-13}$, which is improvement of more than
10 compared with the current bound derived by the MEGA experiment on 98’
Brooks:1999pu , and that a goal of the first phase of the experiment is
$(1$-$2)\times 10^{-13}$ after another two years. Further improvement down to
$\sim 10^{-14}$ in the second phase may be possible after some upgrades.

The signal in $\mu$-$e$ conversion experiments is monochromatic electron with
energy $m_{\mu}-E_{\rm bound}$ ($E_{\rm bound}$ bounding energy), and then the
backgrounds (BGs) are quite suppressed. While the conversion rates are
suppressed by $10^{-(2-3)}$ compared with $Br(\mu\rightarrow e\gamma)$ in
typical BSMs such as supersymmetric models, the high sensitivities to the BSMs
are expected. Two experiments for $\mu$-$e$ conversion searches with
sensitivities $10^{-(16-17)}$, Mu2e in Fermilab mu2e  and COMET in J-parc
comet , are planed. Furthermore, the PRISM/PRIME experiment, in which very
intensive pulsed beam is produced by the FFAG muon storage ring, is also
planed. It has ultimate sensitivity as $10^{-(18-19)}$ comet .

The searches for cLFV in the tau decay will be continued by the Belle
experiment; however, further improvement requires higher luminosity ${\cal
L}$. Now, two experiments, the Super KEKB Akeroyd:2004mj  and the Super flavor
factory Giorgi:2006qj , are planed. The processes $\tau\rightarrow\mu\gamma$
and $\tau\rightarrow e\gamma$ already suffer from irreducible BGs from
$e^{-}e^{+}\rightarrow\tau^{+}\tau^{-}\gamma$, Improvements of the
sensitivities are scaled as $\sqrt{\cal L}$ without reduction of the BGs, and
the reaches are argued $10^{-(8-9)}$. Other cLFV processes in the tau decay
are almost BG-free, and the reaches are argued $10^{-(9-10)}$.

##  II CLFV in BSMs

Following are effective operators with cLFV of muon up to $D=6$,

$\displaystyle{\cal
L}=\frac{m_{\mu}}{\Lambda^{2}}\bar{e}\sigma^{\mu\nu}F_{\mu\nu}\mu+\frac{1}{\Lambda_{F}^{2}}\bar{e}\mu\bar{e}e+\frac{1}{\Lambda_{F}^{2}}\bar{e}\mu\bar{q}q.$
(9)

The first term is for $\mu\rightarrow e\gamma$, and the second and third ones
are for $\mu\rightarrow 3e$ and $\mu$-$e$ conversion in nuclei, respectively.
Here, other LFV terms with different tensor structures are omitted for
simplicity. Roughly speaking, the branching ratios or conversion rates are
$\sim(m_{W}/\Lambda)^{4}$ or $(m_{W}/\Lambda_{F})^{4}$. Thus, $\Lambda$ and
$\Lambda_{F}$ should be larger than $\sim 10^{(5-6)}$ GeV from the
experimental bounds. This implies that the cLFV searches have quite
sensitivities to the BSMs.

We have several motivations to consider the BSMs at TeV scale. The naturalness
argument for the Higgs boson mass terms is one of them. The dark matter in the
universe may be related to physics at TeV scale. In addition, some people
consider origin of the neutrino masses at TeV scale.

Why does nature hide clues of the BSMs from FCNC processes including cLFV
processes? Some of BSMs, such as the two- or multi-Higgs doublet models and
the left-right symmetric models, introduce new Higgs and/or gauge bosons at
TeV scale. Some models, such as supersymmetric models, extra dimension models,
and the little Higgs model with T parity, introduce partners of leptons and
quarks. Those new fields introduce new sources of FCNCs. While FCNCs in the SM
are suppressed by small quark/lepton masses or small mixing angle due to the
GIM mechanism, the suppression is not necessarily automatic in the BSMs.

First, let us consider the cLFV processes in the supersymmetric standard model
(SUSY SM), since it is the leading candidate among the BSMs and also a
prototype of the BSMs. Supersymmetry is a symmetry between bosons and
fermions, and superpartners for each particles in the SM are introduced to the
SUSY SM.

Supersymmetry is not exact in nature. We need to introduce SUSY-breaking mass
terms for SUSY particles, which have not yet been discovered. The SUSY-
breaking terms are new sources of the flavor violation, since the squark and
slepton mass matrices are not necessarily simultaneously diagonalized with
those of quarks and leptons.222 The R parity is assumed here. When it is
violating, new LFV sources are introduced in the SUSY SM. See Ref.
deGouvea:2000cf  for the detail discussion.  This leads to so-called the SUSY
flavor problem. We will discuss this problem from a viewpoint of cLFV in the
following.

In the SUSY SM $\mu\rightarrow e\gamma$ is generated by one-loop diagrams, and
the branching ratio is approximately given as

$\displaystyle{\rm Br}(\mu\rightarrow
e\gamma)\sim\frac{\alpha}{4\pi}\left(\frac{m_{W}}{m_{SUSY}}\right)^{4}\sin^{2}\theta_{\tilde{e}\tilde{\mu}}\left(\frac{\Delta
m_{\tilde{l}}^{2}}{m^{2}_{SUSY}}\right)^{2},$

where $m_{SUSY}$ the SUSY breaking scale, $\sin\theta_{\tilde{e}\tilde{\mu}}$
the slepton mixing angle, ${\Delta m_{\tilde{l}}^{2}}$ is the slepton mass
square difference.

While the branching ratio of $\mu\rightarrow e\gamma$ is suppressed by one-
loop factor, we need much more suppression in order to make it below the
experimental bound. Three directions are proposed. First is the universal
scalar mass hypothesis, in which the squarks and sleptons with common quantum
numbers are degenerate in mass (${\Delta m_{\tilde{l}}^{2}}\ll m_{SUSY}^{2}$).
Many models are constructed following this direction; the gravity mediation
Nilles:1983ge , the gauge mediation gaugemed , the gaugino mediation ginom ,
and the anomaly mediation ams . Second is the alignment hypothesis
($\sin\theta_{\tilde{e}\tilde{\mu}}\ll 1$). It is assumed that squark and
slepton mass matrices can be diagonalized in the same basis as those of quarks
and leptons due to some flavor symmetries or some mechanism Nir:1993mx . Third
is the decoupling hypothesis ($m_{SUSY}\gg m_{W}$). Squarks and sleptons in
the first and second generations are so heavy ($O(10^{4-5})$ GeV) that the
flavor violation in the first and second generations are suppressed effsusy .

In these three hypothesis, the cLFV processes are suppressed. However, the
improvements of the experimental sensitivities may probe the origin of the
SUSY breaking terms and also physics beyond the SUSY SM.

In the universal scalar mass hypothesis, if some physics has LFV interactions
below the SUSY-breaking mediation scale, the LFV slepton mass terms are
induced radiatively by the renormalization-group effect Hall:1985dx . In this
case the LFV mass terms are not suppressed by powers of the energy scale for
the LFV interactions. The seesaw mechanism and the GUTs are nowadays ones of
attractive models from the phenomenological and theoretical points of view. In
these models, LFV Yukawa interactions are introduced. Thus, if the SUSY
breaking mediation scale is higher than the GUT Barbieri:1994pv  or the right-
handed neutrino mass scale Borzumati:1986qx , sizable LFV processes might be
predicted. In Fig. 1, ${\rm Br}(\mu\rightarrow e\gamma)$ in the SUSY seesaw
model is shown. The observed large neutrino mixing angles enhances the cLFV
processes of muon and tau lepton Hisano:1997tc ; Hisano:1998fj . Various
studies are performed under the universal scalar mass hypothesis. See
references in Ref. Raidal:2008jk .

Figure 1: ${\rm Br}(\mu\rightarrow e\gamma)$ in the SUSY seesaw model. This
figure comes from Ref. Hisano:1998fj .

In the decoupling hypothesis, squarks and sleptons in the third generation may
have large flavor violation. In addition, even when SUSY particle masses
larger than $O(1$-$10)$ TeV, the Higgs boson exchange may give sizable
contributions to the cLFV processes Babu:2002et . The SUSY SM has two doublet
Higgs bosons ($H_{1}$ and $H_{2}$), and their effective couplings for leptons
are

$\displaystyle-{\cal
L}_{Y}=\bar{e}_{Ri}Y_{li}L_{i}H_{1}+\bar{e}_{Ri}\Delta_{ij}L_{j}H_{2}^{\dagger}+h.c.,$
(11)

where $Y_{li}$ is the tree-level Yukawa coupling and $\Delta_{ij}$ is non-
holomorphic correction generated at one-loop level. $\Delta_{ij}$ is not
suppressed by the SUSY breaking scale, in addition to enhancement of
$\tan\beta$. Then, when the SUSY particles are heavy enough, the Higgs boson
exchange dominates in the cLFV processes. Fig. 2 shows the branching ratios
for cLFV processes of muon, induced by the Higgs boson exchange in the SUSY
SM. This figure comes from Ref. Paradisi:2006jp . $\mu\rightarrow e\gamma$ is
induced by the Barr-Zee type loop diagrams, while $\mu\rightarrow 3e$ and
$\mu\rightarrow e$ conversion in nuclei are Higgs-boson exchange processes
effectively at tree-level. The Higgs boson exchange contributions to the cLFV
processes of tau lepton are also discussed in Ref. higgstau .

Figure 2: Branching ratios for cLFV processes of muon induced by the Higgs
boson exchange in the SUSY SM. This figure comes from Ref. Paradisi:2006jp .

If the alignment between lepton and slepton masses is not complete in the
third hypothesis, the cLFV processes are predicted. Let us show an example
with $U(1)\times U(1)$ flavor symmetries, which is given in Ref. Feng:2007ke .
The charge assignments of the right- and left-handed leptons are

$\displaystyle\bar{E}_{1}(1,0),~{}\bar{E}_{2}(1,-2),~{}\bar{E}_{3}(0,-3),$
$\displaystyle L_{1}(4,0),~{}L_{2}(2,2),~{}L_{3}(0,4),$

and the flavor symmetries are broken by VEVs of flavon fields,
$\phi_{1}(-1,0)$ and $\phi_{2}(0,-1)$. In this case, the charged lepton mass
matrix is

$\displaystyle(m_{l})\sim\lambda\left(\begin{array}[]{ccc}\lambda^{4}&0&0\\\
\lambda^{4}&\lambda^{2}&0\\\ \lambda^{4}&\lambda^{2}&1\end{array}\right),$
(15)

and the hierarchical structure of lepton masses is explained with $\lambda\sim
0.1-0.2$. In this setup, the slepton mass matrices are aligned, and the left-
handed and right-handed slepton mixings are suppressed as
$\theta_{\tilde{\mu}_{L}\tilde{e}_{L}}\sim\lambda^{4}$ and
$\theta_{\tilde{\mu}_{R}\tilde{e}_{R}}\sim\lambda^{2}$. However, these mixings
are marginal to the bounds from $\mu\rightarrow e\gamma$.

The SUSY flavor problem is one of the guidelines to construct realistic SUSY
SMs. As shown above, in the proposed ideas and models to suppress the FCNC
processes, the cLFV processes are not necessarily suppressed, and the on-going
and planed experiments cover the predictions.

The cLFV processes also have good sensitivities to non-SUSY models at TeV
scale as expected. The BSMs at TeV scale are severely constrained from the
cLFV processes, unless some mechanism works to suppress them. In the
following, some concrete examples are reviewed.

First is the little Higgs model ArkaniHamed:2001ca  with T parity Cheng:2003ju
. In the little Higgs model, the Higgs boson is a pseudo Nambu-Goldstone boson
for symmetry breaking, and quadratic divergence in the radiative correction to
the Higgs boson mass term is cancelled by heavy particles at one-loop level.
The T parity is introduced for the heavy particles not to contribute to the
electroweak observables at tree level. This extension has a bonus. The
lightest T-odd particle is stable and a candidate of the dark matter in the
universe.

In the little Higgs model with T parity, SU(2)L doublet mirror leptons with T
parity odd are introduced. The mirror leptons have coupling with leptons and
the heavy gauge bosons, which is lepton-flavor violating. The cLFV processes
are generated at one-loop level, similar to the SUSY SM. In Fig. 3, ${\rm
Br}(\mu\rightarrow e\gamma)$ and ${\rm Br}(\mu\rightarrow 3e)$ are shown in
this model. It is shown in Ref. Blanke:2007db  that the accidental
cancellation reduces ${\rm Br}(\mu\rightarrow e\gamma)$ and two processes have
comparable branching ratios. Then, this model still is viable even the mirror
leptons are around 1 TeV.

Figure 3: ${\rm Br}(\mu\rightarrow e\gamma)$ and ${\rm Br}(\mu\rightarrow 3e)$
in the little Higgs model with T parity. Here, the mirror leptons masses are
taken from 300 GeV and 1.5 TeV. This figure comes from Ref. Blanke:2007db .

Next is the SM on the Randall-Sundrum (RS) background. The RS geometry is
known as a solution of the hierarchy problem Randall:1999ee . In addition,
when the SM fermions and gauge bosons propagate in the full five-dimensional
space, the fermion mass hierarchy is also explained even from the “anarchic”
structure Gherghetta:2000qt . In this model the Kaluza-Klein (KK) particles
have the LFV interaction. In Fig. 4 ${\rm Br}(\mu\rightarrow e\gamma)$ and
${\rm R}(\mu\rightarrow e;{\rm Ti})$ are shown for the KK scale 10 TeV. The
current experimental bounds give constraints on this model. While
$\mu\rightarrow e\gamma$ is a one-loop process, the $\mu\rightarrow e$
conversion is generated at tree level.

Figure 4:  ${\rm Br}(\mu\rightarrow e\gamma)$ and ${\rm R}(\mu\rightarrow
e;{\rm Ti})$ in the SM on the Randall-Sundrum background. Here, the Kaluza-
Klein scale is 10 TeV. This figure comes from Ref. Agashe:2006iy .

Finally, we discuss the relation of the cLFV processes and the neutrino
masses. The seesaw mechanism is the most promising in the candidates of the
neutrino masses. In many models the mechanism is realized at much higher
energy scale than the TeV scale. In such models we cannot expect some direct
relations between the cLFV processes and the origin of the neutrino masses.
Exceptions are the SUSY seesaw models susyseesaw1 ; susyseesaw2 ; however, the
relations are indirect. On the other hand, there are attempts to construct
models of the neutrino mass origin at TeV scale.

One of the models is the triplet Higgs model (type-II seesaw model)
Schechter:1980gr . In this model, the triplet Higgs boson with mass at TeV
scale has sizable lepton-flavor violating coupling, and the coupling is
directly linked to the neutrino mass matrix. In Fig. 5 the branching ratios
for the cLFV processes of muon in this model are shown. The pattern depends on
the neutrino mass structure Kakizaki:2003jk .

Figure 5: Branching ratios for cLFV processes of muon in the triplet Higgs
model. Here, neutrino mass spectrum is inverted-hierarchical (hierarchical) in
the upper (lower) figure. These figures come from Ref. Kakizaki:2003jk

##  III Correlations

When the cLFV processes are discovered, it would be important to take
correlations among various processes in order to unveil the origin of the
lepton-flavor violation and the realistic BSM.

The first correlation is among the cLFV processes with common flavor
transition. The correlation discriminates BSMs at TeV scale since the pattern
of correlation depends on models. In Table 1, the ratios of the cLFV rates of
muon are shown in three models; the SUSY SM in which the dipole moment
contribution to the cLFV processes is dominant, the SUSY SM in which the SUSY
particles are so heavy that the Higgs boson exchange dominates, and the little
Higgs model with T parity. While the cLFV processes are radiatively generated
in those models, the pattern of the ratio of the cLFV rates are quite
different. Correlations among the cLFV processes of $\tau$ and among the
$\mu$-$e$ conversions in various nuclei Kitano:2002mt  are also useful to
discriminate models.

Table 1: Ratios of the cLFV rates of muon in three models. This table comes from Blanke:2007db .  | $\frac{{\rm Br}(\mu\rightarrow 3e)}{{\rm Br}(\mu\rightarrow e\gamma)}$ | $\frac{{\rm R}(\mu\rightarrow e;{\rm Ti})}{{\rm Br}(\mu\rightarrow e\gamma)}$
---|---|---
SUSY SM (dipole) | $\sim 6\times 10^{-3}$ | $\sim 5\times 10^{-3}$
SUSY SM (Higgs) | $\sim 6\times 10^{-3}$ |  $0.08$-$0.15$
Little Higgs with T parity | 0.4-2.5 |  $10^{-2}$-$10^{2}$

The second correlation is among the cLFV processes with different flavor
transitions. In some models the cLFV processes are related to the neutrino
masses. In the triplet Higgs model, the interaction of the triplet Higgs boson
with leptons is directly linked to the neutrino mass matrix $(m_{\nu})$, as
mentioned above. The model is one of realizations of the minimal flavor
violation hypothesis in the lepton sector Cirigliano:2005ck . As the result,
the ratios of the cLFV rates of tau lepton and muon are given as

$\displaystyle\frac{{\rm Br}(\tau\rightarrow\mu\gamma)}{{\rm
Br}(\mu\rightarrow e\gamma)}\simeq
0.17\times\left(\frac{(m_{\nu}m_{\nu}^{\dagger})_{\tau\mu}}{(m_{\nu}m_{\nu}^{\dagger})_{\tau
e}}\right)^{2}\sim 300$ $\displaystyle\frac{{\rm Br}(\tau\rightarrow
e\gamma)}{{\rm Br}(\mu\rightarrow e\gamma)}\simeq
0.17\times\left(\frac{(m_{\nu}m_{\nu}^{\dagger})_{\tau
e}}{(m_{\nu}m_{\nu}^{\dagger})_{\tau e}}\right)^{2}\sim 0.2.$ (16)

In the SUSY seesaw models, the cLFV processes are indirectly related to the
origin of the neutrino masses even if the energy scale is high. Under the
universal scalar mass hypothesis, the cLFV processes give information of the
type-I SUSY seesaw model, which is independent of the neutrino mass from a
viewpoint of the reconstruction of the model susyseesaw2 . In the type-II SUSY
seesaw model, the relation in Eq. 16 is predicted again susyseesaw2 .

The third correlation is between the hadronic and leptonic FCNC processes. In
GUTs, quarks and leptons are unified so that the cLFV processes are correlated
with hadronic FCNC processes Hisano:2003bd . Let us consider the SUSY SU(5)
GUT with right-handed neutrinos. In this model, lepton doublets and right-
handed down-type quarks are embedded in common SU(5) multiplets. Then,
neutrino Yukawa coupling affects both the right-handed sdown and left-handed
slepton mass matrices Moroi:2000mr .

In Fig. 6 we show the correlation between CP phase of the $B_{s}$ mixing
amplitude, $\phi_{B_{s}}$, and ${\rm Br}(\tau\rightarrow\mu\gamma)$ in the
SUSY SU(5) GUT with right-handed neutrinos. The CP violation in the $B_{s}$
mixing is suppressed in the SM, and then it is also sensitive to the BSMs. The
phase $\phi_{B_{s}}$ is defined to be zero in the SM. It is recently announced
by the Utfit collaboration Bona:2008jn  that $\phi_{B_{s}}$ deviates more than
$3\sigma$ from the SM prediction. In the figure, we show the region for
$\phi_{B_{s}}$. The deviation of $\phi_{B_{s}}$ is constrained from null
result of $\tau\rightarrow\mu\gamma$ search in this model, and the $95\%$
probability region derived by the Utfit collaboration is marginal in this
model. When the deviation of $\phi_{B_{s}}$ is established, search for
$\tau\rightarrow\mu\gamma$ would be an important test of the SUSY SU(5) GUTs.

Figure 6:  Correlation between CP phase of the $B_{s}$ mixing amplitude,
$\phi_{B_{s}}$, and ${\rm Br}(\tau\rightarrow\mu\gamma)$ in the SUSY SU(5) GUT
with right-handed neutrinos. Here, $\tan\beta=10$ and 30, and we show the
region for $\phi_{B_{s}}$, derived by the Utfit collaboration. These figures
come from Ref. Hisano:2008df

.

When the flavor-violating mass terms for the right-handed squarks are non-
vanishing, the hadronic EDMs are generated at one-loop Dimopoulos:1994gj  and
two-loop levels Hisano:2007cz . The non-zero $(m_{\tilde{d}_{R}}^{2})_{23}$
generates the strange-quark chromoelectric dipole moment, which contributes to
the hadronic EDMs Hisano:2003iw . Thus, the correlation between the hadronic
EDMs and the cLFV processes would be the tests of the SUSY GUTs. But, this
program still has difficulties in precisions of the hadronic EDM evaluation
Narison:2008jp , and the further improvements would be required for it.

##  IV Summary

In this talk, I discussed charged lepton-flavor violation in physics beyond
the standard model and reviewed topics related to it. The cLFV processes are
accidentally suppressed by the GIM mechanism in the SM even after tiny
neutrino masses are introduced. On the other hand, the suppression is not
necessarily automatic in physics beyond the SM. Studies of cLFVs probe BSMs,
hidden flavor symmetries, and underlying flavor structures.

Current bounds on cLFVs give constraints on physics even around
O($10^{(5-6)}$) GeV. In practical BSMs, cLFVs are suppressed by loop-factors
or small flavor-mixing, or accidental cancellation. Thus, coming MEG
experiment, and planed experiments, Mu2e, COMET and PRISM/PRIME, will cover
interesting regions in various BSMs. Since the cLFVs are pieces of puzzles in
the BSMs, it is important to stress that taking various correlations are
useful to solve the puzzles.

###### Acknowledgements.

The work of JH was also supported in part by the Grant-in-Aid for Science
Research, Japan Society for the Promotion of Science (No. 20244037 and No.
2054252).

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