# Some Examples of Bicrossed Products with the Rapid Decay Property

Hua Wang

###### Abstract

We consider bicrossed products obtained by twisting compact semi-direct
products by a suitable finite subgroup. Under some restriction, we give a
practical criterion for the discrete dual of such bicrossed products to have
the rapid decay property (property (RD)). Using this theory, we construct some
examples of discrete quantum groups with (RD) but do not grow polynomially.
Further examples that do not satisfy the hypothesis of our main result are
also constructed.

##  1 Introduction

The rapid decay property, which shall be referred as property ${\rm(RD)}$, or
simply ${\rm(RD)}$ in this paper, be it for discrete groups (Jolissaint
[MR943303]) or discrete quantum groups (Vergnioux [MR2329000], see also
Bhowmick, Voigt & Zacharias [MR3448333] for a non-unimodular variation), is an
interesting approximation property, partly due to its connection with
$K$-theory, $L^{2}$-homology et cetera. On the other hand, from its
introduction by Kac ([MR0229061]) in the 1960s to the culmination work of Vaes
& Vainerman ([MR1970242]) in the general setting of locally compact groups,
the bicrossed product construction, as a systematic procedure to produce large
classes of non-commutative and non-cocommutative quantum groups, plays an
important role in the development of the still rapidly growing field of
quantum groups. In the work of Fima & the author [MR4345210], we give a
systematic theoretical study of property ${\rm(RD)}$ for (the discrete dual
of) bicrossed products $\Gamma\bowtie G$ of a matched pair $(\Gamma,G)$ of a
compact group $G$ with a discrete one $\Gamma$. The main results there are
that $\widehat{\Gamma\bowtie G}$ has ${\rm(RD)}$ if and only if both $\Gamma$
and $\widehat{G}$ have ${\rm(RD)}$ in a compatible manner (Theorem 2.8 below
has a more precise formulation). One key component in formulating these
results is the notion of a matched pair of length functions, which is
intimately related to the fusion structure of irreducible representations of
the compact quantum group $\Gamma\bowtie G$. We point out that one major
drawback of these theoretical results is that, in practice, due to the
complexity of this notion, it can be quite difficult to verify that the
witnessing length functions for the relevant approximation properties like
${\rm(RD)}$ or polynomial growth are matched or not. This is the main reason
for the lack of concrete examples in [MR4345210].

This paper picks up where [MR4345210] left off, namely, the construction of
examples, to support the theory developed there. We work in a particular class
of bicrossed products already considered in Fima, Mukherjee & Patri
[MR3743231], which we now briefly describe and refer to § 3 for more details.
We start from a nontrivial semidirect product $G\rtimes\Gamma$ with $G$ a
compact group and $\Gamma$ a discrete one, as well as a finite subgroup
$\Lambda$ of $\Gamma$ that does not lie entirely in the center of $\Gamma$.
One can then show that the pair $(\Gamma,G\rtimes\Lambda)$ is matched in a
natural way (again see § 3), where the underlying left action
$\alpha^{\Lambda}:\Gamma\curvearrowright G\rtimes\Lambda$ and right action
$\beta^{\Lambda}:\Gamma\curvearrowleft G\rtimes\Lambda$ are both nontrivial,
hence yields a non-commutative, non-cocommutative bicrossed product
$\Gamma\bowtie(G\rtimes\Lambda)$. We restrict our attention to the study of
${\rm(RD)}$ and the closely related notion of polynomial growth of the dual
$\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$. This has the advantage that the
underlying representation theory can be divided into two well-understood
parts: the first part is the representation theory of the bicrossed products
studied in [MR4345210], which, modulo the relations determined by the matched
pair $(\Gamma,G\rtimes\Lambda)$, reduces to the second part concerning the
representation theory of the semidirect product $G\rtimes\Lambda$, which is a
special case studied in [wang2019representations]. Moreover, the finiteness of
the group $\Lambda$ provides a way to control the length functions, which
eventually leads to a characterization result that gets rid of the technically
involved notion of matched pair of length functions (see Definition 2.6), this
is the contents of Theorem 1.2.

We now state the main theoretical results of this paper consisting of the
following two theorems.

######  Theorem 1.1.

In the above settings. If there is a $\Gamma$-invariant length function
$l_{\widehat{G}}$ on $\widehat{G}$, and a $\beta^{\Lambda}$-invariant length
function $l_{\Gamma}$ on $\Gamma$, such that both $\left\lparen\mskip
1.0mu\widehat{G}\mskip 2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$ and
$\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0mul_{\Gamma}\,\right\rparen$ have polynomial growth (resp. $(RD)$), then
the dual of the bicrossed product, namely
$\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$, also has polynomial growth (resp.
$(RD)$).

######  Theorem 1.2.

Let $\widetilde{\tau}:\Gamma\to\operatorname{Out}(G)$ be the composition of
$\tau:\Gamma\to\operatorname{Aut}(G)$ with the canonical projection
$\operatorname{Aut}(G)\to\operatorname{Out}(G)$. If
$\operatorname{Image}(\widetilde{\tau})$ is finite, then the following are
equivalent:

  1. (1)

$\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$ has polynomial growth (resp.
$(RD)$);

  2. (2)

both $\Gamma$ and $\widehat{G}$ have polynomial growth (resp. $(RD)$).

Theorem 1.1 provides a sufficient condition for
$\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$ to have ${\rm(RD)}$ (resp.
polynomial growth). Albeit still non-trivial, this condition is much easier to
check in some concrete cases than the condition of matched pair of length
functions used in Theorem 2.7 and Theorem 2.8, which we shall exploit in § 8.

Theorem 1.2 is a stronger result, provided that the assumption there is
satisfied, as it completely gets rid of the difficult problem of checking the
compatibility of the relevant length functions. This leads to some concrete
examples of $\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$ having ${\rm(RD)}$ but
not polynomial growth (§ 7). However, it has the limitation of not covering
many other interesting examples, as will be demonstrated again in § 8. It is
in the opinion of the author that a more unified approach is desired that
relaxes the assumption of Theorem 1.2 while still at least includes the
examples in § 8.

This paper is organized as follows. In the long preliminary § 2, we provide
all the needed background on the relevant representation theory, which can be
rather involved. Besides the purpose of making the paper reasonably self-
contained, it also serves to fix the notation for our later use. In § 3, we
briefly give a conceptual and still elementary explanation of how one can
obtain non-trivial matched pair by twisting semidirect products by a finite
subgroup, followed by § 4 which fixes more notation to help us deal with the
arising subtleties. After the stage is set, we prove the main theorems in § 5
and § 6. Finally, using these tools, in § 7, we illustrate how to construct
the our desired examples using Theorem 1.2, and further examples are
considered in § 8, showing the limitation of the procedure in § 7.

We finish this introduction by making some convention.

###  1.1 Convention and notation

Regarding the terminologies and notation of compact quantum groups, our
choices are mainly in consistent with the works of Neshveyev & Tuset
[MR3204665] and Woronowicz [MR1616348].

All representations and projective representations of (quantum) groups are
assumed to be unitary and finite dimensional unless otherwise stated. We view
discrete quantum groups as the dual of compact ones.

When $\mathbb{G}$ is a compact (quantum) group, the set of equivalence classes
of irreducible representations of $\mathbb{G}$ is denoted by
$\operatorname{Irr}(\mathbb{G})$, and the trivial representation of
$\mathbb{G}$ by $\varepsilon_{\mathbb{G}}$, or simply by $\varepsilon$ when
$\mathbb{G}$ is clear from context. If $u$ is a representation of
$\mathbb{G}$, then the equivalence class of $u$ is denoted by $[u]$. The dual
of $\mathbb{G}$, as a discrete quantum group, is denoted by
$\widehat{\mathbb{G}}$. The tensor product of two representations $u$ and $v$
of $\mathbb{G}$ is denoted by $u\times v$, and the tensor product of classes
$x,y\in\operatorname{Irr}(\mathbb{G})$ by $x\otimes y$. The space of operators
that intertwine $u$ and $v$ is denoted by
$\operatorname{Mor}_{\mathbb{G}}(u,v)$.

When $H$, $K$ are Hilbert spaces, the notation $B(H,K)$ means the Banach space
of all bounded linear operators from $H$ to $K$. The space $B(H,H)$ is denoted
by $B(H)$.

For a topological group $G$, we use $\operatorname{Inn}(G)$ to denote the
normal subgroup of $\operatorname{Aut}(G)$ consisting of inner automorphisms,
where $\operatorname{Aut}(G)$ is the group of topological automorphisms of
$G$. The outer automorphism group $\operatorname{Out}(G)$ is of course the
quotient group $\operatorname{Aut}(G)/\operatorname{Inn}(G)$.

###  1.2 Acknowledgement

This work is supported by the ANR project ANCG (No. ANR-19-CE40-0002). The
author would also like to thank Professor Pierre Fima for several useful
discussions.

##  2 Preliminaries

As preliminaries, we briefly describe the relevant results of ([MR4345210])
and ([wang2019representations]), which form the theoretical framework of this
paper. To make the presentation more self-contained, we also briefly recall
the bicrossed product construction as presented in [MR3743231], and property
${\rm(RD)}$ in the discrete quantum group setting ([MR2329000], see also
[MR3448333]).

###  2.1 Rapid decay and polynomial growth for discrete quantum groups of Kac
type

As the bicrossed products of matched pair of groups considered here are always
of Kac type, we only recall the relevant formulation in this simplified
setting.

Let $\mathbb{H}$ be a compact quantum group of Kac type and
$\widehat{\mathbb{H}}$ its dual (as a discrete quantum group of Kac type),
$\ell^{\infty}(\widehat{\mathbb{H}})$ be the $\ell^{\infty}$-direct sum
$\bigoplus_{x\in\operatorname{Irr}(\mathbb{H})}^{\ell^{\infty}}B(H_{x})$, and
$c_{c}(\widehat{\mathbb{H}})$ the ideal of the algebraic direct sum, whose
$C^{\ast}$-completion is denoted by $c_{0}(\widehat{\mathbb{H}})$. A length
function on $\widehat{\mathbb{H}}$ is a map
$l:\operatorname{Irr}(\mathbb{H})\to\mathbb{R}_{\geq 0}$ such that  (i)
$l([\varepsilon_{\mathbb{H}}])=0$;  (ii) $l(x)=l(\overline{x})$for all
$x\in\operatorname{Irr}(\mathbb{H})$;  (iii) $l(z)\leq l(x)+l(y)$if
$z\subseteq x\otimes y$.  In the following, $p_{x}$ denotes the central
projection of $c_{0}(\widehat{\mathbb{H}})$ that corresponds to the block
$B(H_{x})$. Take $a\in c_{c}(\widehat{\mathbb{H}})$, the Fourier transform
$\mathcal{F}_{\mathbb{H}}(a)$ of $a$ is defined to be

$\mathcal{F}_{\mathbb{H}}(a)=\sum_{x\in\operatorname{Irr}(\mathbb{H})}(\dim
x)\left[(\operatorname{Tr}_{H_{x}}\otimes\operatorname{id})(u^{x}(ap_{x}\otimes
1))\right]\in\operatorname{Pol}(\mathbb{H})$

where $u^{x}$ is a unitary representation on $H_{x}$, chosen (and fixed) in
the class $x$. The Sobolev-$0$-norm $\left\|a\right\|_{\mathbb{H},0}$ is
determined by

$\left\|a\right\|_{\mathbb{H},0}^{2}=\sum_{x\in\operatorname{Irr}(\mathbb{H})}(\dim
x)\operatorname{Tr}_{H_{x}}(a^{\ast}ap_{x}).$

Given a length function $l$ on $\widehat{\mathbb{H}}$, consider the element
$L=\sum_{x\in\operatorname{Irr}(\mathbb{H})}l(x)p_{x}$ that is affiliated (in
the sense of Woronowicz [MR1096123]) to $c_{0}(\widehat{\mathbb{H}})$. Denote
the spectral projection of $L$ corresponding to the interval
$\interval[openright]{n}{n+1}$ by $q_{n}$. The pair $\left\lparen\mskip
1.0mu\widehat{\mathbb{H}}\mskip 2.0mu,\mskip 3.0mul\,\right\rparen$ is said to
have

  * •

_the rapid decay property_ or property ${\rm(RD)}$, if there exists a
polynomial $P\in\mathbb{R}[X]$, such that for all $k\in\mathbb{N}$ and $a\in
q_{k}c_{c}(\widehat{\mathbb{H}})$, we have
$\left\|a\right\|_{C(\mathbb{H})}\leq P(k)\left\|a\right\|_{\mathbb{H},0}$,
where $C(\mathbb{H})$ is the $C^{\ast}$-algebra of the reduced version of
$\mathbb{H}$;

  * •

_polynomial growth_ , if there exists a polynomial $P\in\mathbb{R}[X]$, such
that for all $k\in\mathbb{N}$,

$\sum_{x\in\operatorname{Irr}(\mathbb{H}),\,k\leq l(x)<k+1}{(\dim x)}^{2}\leq
P(k).$

We say that $\widehat{\mathbb{H}}$ has ${\rm(RD)}$ (resp. polynomial growth),
if there exists a length function $l$ on $\widehat{\mathbb{H}}$ such that the
pair $\left\lparen\mskip 1.0mu\widehat{\mathbb{H}}\mskip 2.0mu,\mskip
3.0mul\,\right\rparen$ has ${\rm(RD)}$ (resp. polynomial growth).

From the work [MR2329000], it is known that the pair $\left\lparen\mskip
1.0mu\widehat{\mathbb{H}}\mskip 2.0mu,\mskip 3.0mul\,\right\rparen$ having
polynomial growth implies it having ${\rm(RD)}$, and the converse holds if
$\mathbb{H}$ is co-amenable; it is also shown there that the duals of compact
connected real Lie groups have polynomial growth.

###  2.2 Bicrossed products and ${\rm(RD)}$

####  2.2.1 Construction of the bicrossed product

We briefly describe the bicrossed product construction associated to a matched
pair of classical groups, which is the type of examples to be considered in
this work, and refer to [MR3743231] or [phdthesis, Chapter I] for details and
more background.

A matched pair of groups $\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0muG\,\right\rparen$ consists of a discrete group $\Gamma$, a compact
(compact means quasi-compact and Hausdorff in this paper) group $G$, together
with two actions $\alpha$ and $\beta$, where $\alpha:\Gamma\curvearrowright G$
is a right action of the group $\Gamma$ on the underlying set of $G$, and
$\beta:\Gamma\curvearrowleft G$ a right action of the group $G$ on the
underlying set of $\Gamma$, such that

$\gamma\in\Gamma,g,h\in
G,\implies\alpha_{\gamma}(gh)=\alpha_{\gamma}(g)\alpha_{\beta_{g}(\gamma)}(h),$
(2.1a) $r,s\in\Gamma,g\in
G,\implies\beta_{g}(rs)=\beta_{\alpha_{s}(g)}(r)\beta_{g}(s),$ (2.1b)
$\gamma\in\Gamma,g\in
G,\implies\alpha_{\gamma}(e_{G})=e_{G},\;\beta_{g}(e_{\Gamma})=e_{\Gamma},$
(2.1c)

where $e_{\Gamma}$ (resp. $e_{G}$) denotes the neutral element in the group
$\Gamma$ (resp. $G$). It is well-known that $(\Gamma,G)$ is a matched pair of
groups if and only if $\Gamma$ and $G$ can be simultaneously identified with
(topological) subgroups of a locally compact group $H$ in such a way that
$H=\Gamma G$, $\Gamma\cap G=\left\\{e_{H}\right\\}$. In this identification,
the underlying actions $\alpha$, $\beta$ are characterized by the relation
$\gamma g=\alpha_{\gamma}(g)\beta_{g}(\gamma)$. One also proves that the
actions $\alpha$, $\beta$ are automatically continuous, and $\alpha$ preserves
the Haar measure of $G$.

Given a matched pair of groups $\left\lparen\mskip 1.0mu\Gamma\mskip
2.0mu,\mskip 3.0muG\,\right\rparen$ with underlying actions $\alpha$ and
$\beta$. To simplify notation, we often denote the right action $\beta$ by a
dot, thus writing $\gamma\cdot g$ instead of $\beta_{g}(\gamma)$. In this
spirit, the orbit of $\gamma\in\Gamma$ under $\beta$ is written as
$\gamma\cdot G$. By the continuity of the action $\beta$, we see that the set
$G_{r,s}:=\left\\{g\in G\;\middle|\;r\cdot g=s\right\\}$ is clopen, thus its
characteristic function $v_{r,s}$ is an element of $C(G)$. It is obvious that
$G_{r,s}\neq\emptyset$ if and only if $r,s$ belong to the same orbit, and
$G_{\gamma}:=G_{\gamma,\gamma}$ is the isotropy subgroup of $G$ fixing
$\gamma\in\Gamma$. As $G_{\gamma}$ is of finite index in $G$ (since $G$
compact and $G_{\gamma}$ open), each $\beta$-orbit is finite, and we denote
the set of all $\beta$-orbits by $\operatorname{Orb}_{\beta}$.

The action $\alpha:\Gamma\curvearrowright G$ induces a $C^{\ast}$-dynamical
system $(C(G),\Gamma,\widetilde{\alpha})$, where $C(G)$ is the algebra of
continuous functions on $G$, and
$\widetilde{\alpha}:\Gamma\to\operatorname{Aut}(C(G))$ the group homomorphism
$\gamma\mapsto\left\\{\alpha_{\gamma^{-1}}^{\ast}:\varphi\mapsto\varphi\circ\alpha_{\gamma^{-1}}\right\\}$.
On the reduced crossed product $\Gamma\rtimes C(G)$, there exists a unique
compact quantum group structure whose comultiplication $\Delta$ satisfies the
following conditions:

  * •

$\Delta$ restricts to the comultiplication on $C(G)$ that is induced by the
multiplication on $G$;

  * •

$\Delta(u_{\gamma})=\sum_{\mu\in\gamma\cdot G}u_{\gamma}v_{\gamma,\mu}\otimes
u_{\mu}$, where we use $u_{r}$ to denote the copy of $r\in\Gamma$ in the
crossed product $\Gamma\rtimes C(G)$.

We denote the compact quantum group $(\Gamma\rtimes C(G),\Delta)$ by
$\Gamma\bowtie_{\alpha,\beta}G$, or simply $\Gamma\bowtie G$, dropping the
actions $\alpha$ and $\beta$ when the context makes them clear, and call this
compact quantum group the bicrossed product of the matched pair of groups
$\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip 3.0muG\,\right\rparen$. The
quantum group $\Gamma\bowtie G$ is always of Kac type, so § 2.1 applies to
$\widehat{\Gamma\bowtie G}$.

Throughout the rest of § 2.2, we fix a matched pair of groups
$\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip 3.0muG\,\right\rparen$. We
denote the underlying left action by $\alpha:\Gamma\curvearrowright G$, and
the right action by $\beta:\Gamma\curvearrowleft G$.

####  2.2.2 Representation theory of the bicrossed product

The representation theory of $\Gamma\bowtie G$ that we will need includes a
classification of irreducible representations, the description of the
conjugate representation (which is the contragredient representation since
$\Gamma\bowtie G$ is of Kac type) in terms of this classification, and most
importantly, the fusion rules. The results summarized here are first obtained
in [MR4345210], with the exception of the simplified version of the fusion
rules, Theorem 2.4, which is given in [phdthesis, Theorem I.4.9]. We refer the
reader to [MR4345210, §3] and [phdthesis, §I.2 & §I.4] for complete proofs of
the results below. Motivational considerations of the classification of
irreducible representations can be found in [phdthesis, §I.3].

We will formulate our results using the notion of
$\mathscr{O}$-representations, which we now briefly describe. Take an orbit
$\mathscr{O}\in\operatorname{Orb}_{\beta}$. By an $\mathscr{O}$-representation
of $G$, we mean a unitary operator $U$ in $B(\ell^{2}(\mathscr{O}))\otimes
B(H)\otimes C(G)$, where $H$ is some finite dimensional Hilbert space, such
that if we write

$U=\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes u_{r,s}$

with ${(e_{r,s})}$ the canonical matrix units of $B(\ell^{2}(\mathscr{O}))$
and $u_{r,s}\in B(H)\otimes C(G)$, then (recall that $v_{r,s}$ is the
characteristic function of $G_{r,s}$)

$\forall
r,s\in\mathscr{O},\;u^{\ast}_{r,s}u_{r,s}=u_{r,s}u^{\ast}_{r,s}=\operatorname{id}_{H}\otimes
v_{r,s}\in B(H)\otimes C(G).$

Viewing each $u_{r,s}$ as a map from $G$ into $B(H)$, then the restriction
$u_{r,r}|_{G_{r}}$ is a unitary representation of $G_{r}$ on $H$. We say that
$U$ is $\mathscr{O}$-irreducible if $u_{r,r}|_{G_{r}}$ is irreducible for some
$r\in\mathscr{O}$, which is equivalent to $u_{r,r}|_{G_{r}}$ being irreducible
for all $r\in\mathscr{O}$. If $W=\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes
w_{r,s}$ is another $\mathscr{O}$-representation of $G$, say on
$\ell^{2}(\mathscr{O})\otimes K$, we say that $U$ and $W$ are
$\mathscr{O}$-equivalent if there exists a unitary operator $T\in B(H,K)$ such
that $T$ intertwines $u_{r,r}|_{G_{r}}$ and $v_{r,r}|_{G_{r}}$ for some
$r\in\mathscr{O}$, which is also equivalent to $T$ intertwines
$u_{r,r}|_{G_{r}}$ and $v_{r,r}|_{G_{r}}$ for all $r\in\mathscr{O}$.

The $\mathscr{O}$-representations can be used to produce representations of
the bicrossed product $\Gamma\bowtie G$ in the following way. If $U$ is an
$\mathscr{O}$-representation as above, then the unitary operator

$\mathfrak{R}_{\mathscr{O}}(U):=\sum_{\gamma\in\mathscr{O}}\left(\sum_{\gamma\in\mathscr{O}}e_{\gamma,\gamma}\otimes\operatorname{id}_{\mathscr{H}}\otimes
u_{\gamma}\right)U$ (2.2)

is a unitary representation of $\Gamma\bowtie G$ on
$\ell^{2}(\mathscr{O})\otimes H$. For
$\mathscr{O}\in\operatorname{Orb}_{\beta}$, denote the set of
$\mathscr{O}$-equivalence classes of $\mathscr{O}$-irreducible representations
by $\operatorname{Irr}_{\mathscr{O}}(\Gamma\bowtie G)$.

######  Theorem 2.1 (Classification of irreducible representations).

The map

$\begin{split}\mathfrak{R}:\coprod_{\mathscr{O}\in\operatorname{Orb}_{\beta}}\operatorname{Irr}_{\mathscr{O}}(G)&\to\operatorname{Irr}(\Gamma\bowtie
G)\\\
[U]\in\operatorname{Irr}_{\mathscr{O}}(G)&\mapsto[\mathfrak{R}_{\mathscr{O}}(U)]\end{split}$
(2.3)

is a well-defined bijection.

The $\mathscr{O}$-irreducible representations are in fact just suitable copies
of induced representations of irreducible representations of $G_{\gamma}$ for
any $\gamma\in\mathscr{O}$. The precise formulation of this correspondence is
given by the following proposition.

######  Proposition 2.2.

Let $\mathscr{O}$ be a $\beta$-orbit, $\gamma\in\mathscr{O}$, and
$u:G_{\gamma}\to\mathcal{B}(\mathscr{H})$ a finite dimensional unitary
representation of $G_{\gamma}$. Take $\sigma_{\mu}\in G_{\gamma,\mu}$ for each
$\mu\in\mathscr{O}$ with $\sigma_{\gamma}=e_{G}$ and define

$u_{r,s}(g)=\begin{cases}u\left(\sigma_{r}g\sigma_{s}^{-1}\right),&\text{if
}g\in G_{r,s};\\\ 0,&\text{if }g\notin G_{r,s}.\end{cases}$

Then the operator $U=\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes u_{r,s}$ is an
$\mathscr{O}$-representation on $\ell^{2}(\mathscr{O})\otimes\mathscr{H}$, and
$U\simeq\operatorname{Ind}_{G_{\gamma}}^{G}(u)$.

Moreover, for any $\gamma\in\mathscr{O}$, the map

$\begin{split}\varPhi_{\gamma}:\operatorname{Irr}(G_{\gamma})&\to\operatorname{Irr}_{\mathscr{O}}(G)\\\
[u]&\mapsto\left[\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes
u_{r,s}\right],\end{split}$ (2.4)

where $u_{r,s}$ is defined above, is a well-defined (i.e. independent of the
representative $u$ of the class $[u]$ and independent of the choices of
$\sigma_{\mu}\in G_{\gamma,\mu}$) bijection.

For an $\mathscr{O}$-representation $U=\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes
u_{r,s}$ on $\ell^{2}(\mathscr{O})\otimes H$, one can show that in fact
$u_{r,s}\in B(H)\otimes\operatorname{Pol}(G)$. Let $j:B(H)\to B(\overline{H})$
be the canonical conjugate linear anti-isomorphism, where $\overline{H}$ is
the conjugate Hilbert space of $H$, and let $S$ be the antipode of the
Hopf-$\ast$-algebra $\operatorname{Pol}(G)$. The matched pair relations
(2.1a), (2.1b) and (2.1c) imply that
$\mathscr{O}^{-1}=\left\\{\gamma^{-1}\;\middle|\;\gamma\in\mathscr{O}\right\\}$
is also a $\beta$-orbit. The unitary

$U^{{\dagger}}:=\sum_{r,s\in\mathscr{O}}e_{s^{-1},r^{-1}}\otimes(\operatorname{id}\otimes\alpha_{s^{-1}}^{\ast})\left[(j\otimes
S)u_{r,s}\right]\in B(\ell^{2}(\mathscr{O}^{-1}))\otimes
B(\overline{H})\otimes C(G)$

is an $\mathscr{O}^{-1}$ representation of $G$ on
$\ell^{2}(\mathscr{O}^{-1})\otimes\overline{H}$, which is equivalent to the
conjugate representation of $U$.

######  Theorem 2.3 (Conjugate of irreducible representations).

The operation $U\mapsto U^{{\dagger}}$ preserves $\mathscr{O}$-equivalence
classes, and induces a well defined involution on
$\coprod_{\mathscr{O}\in\operatorname{Orb}_{\beta}}\operatorname{Irr}_{\mathscr{O}}(G)$,
and the classification map
$\mathfrak{R}:\coprod_{\mathscr{O}\in\operatorname{Orb}_{\beta}}\operatorname{Irr}_{\mathscr{O}}(G)\to\operatorname{Irr}(\Gamma\bowtie
G)$ preserves involution, where the involution on the latter set is given by
conjugation. Moreover, if $\gamma\in\mathscr{O}\in\operatorname{Orb}_{\beta}$,
and $U$ is an $\mathscr{O}$-representation such that
$U\simeq\operatorname{Ind}(u)$ for some irreducible representation $u$ of
$G_{\gamma}$, then $\alpha_{\gamma^{-1}}$ restricts to an isomorphism of
compact groups from $G_{\gamma^{-1}}$ onto $G_{\gamma}$, and
$U^{{\dagger}}\simeq\operatorname{Ind}_{G_{\gamma^{-1}}}^{G}(u\circ\alpha_{\gamma^{-1}})$.

To describe the fusion rules of $\Gamma\bowtie G$, we need a certain twisted
tensor product which we now describe. For $i=1,2,3$, let
$\mathscr{O}_{i}\in\operatorname{Orb}_{\beta}$, suppose
$U_{i}=\sum_{r,s\in\mathscr{O}_{i}}e_{r,s}\otimes u_{r,s}^{(i)}$ is an
$\mathscr{O}_{i}$-representation on $\ell^{2}(\mathscr{O}_{i})\otimes H_{i}$
and $W_{i}=\mathfrak{R}_{\mathscr{O}_{i}}(U_{i})$. We will describe
$\dim\operatorname{Mor}_{\Gamma\bowtie G}(W_{3},W_{1}\times W_{2})$, which is
slightly more general than the fusion rules (for which we only need the case
where $U_{i}$ is $\mathscr{O}_{i}$-irreducible for all $i$).

In $\Gamma$, the product set

$\mathscr{O}_{1}\mathscr{O}_{2}=\left\\{\gamma_{1}\gamma_{2}\;\middle|\;(\gamma_{1},\gamma_{2})\in\mathscr{O}_{1}\times\mathscr{O}_{2}\right\\}$

is a disjoint union of $\beta$-orbits. For
$\gamma\in\mathscr{O}_{1}\mathscr{O}_{2}$, put

$K^{\gamma}_{\mathscr{O}_{1},\mathscr{O}_{2}}=\operatorname{Vect}\left\\{\delta_{\gamma_{1}}\otimes\delta_{\gamma_{2}}\;\middle|\;(\gamma_{1},\gamma_{2})\in\mathscr{O}_{1}\times\mathscr{O}_{2}\text{
with
}\gamma_{1}\gamma_{2}=\gamma\right\\}\subseteq\ell^{2}(\mathscr{O}_{1})\otimes\ell^{2}(\mathscr{O}_{2}),$

and define a map $U_{1}\times_{\gamma}U_{2}:G_{\gamma}\to
B(K^{\gamma}_{\mathscr{O}_{1},\mathscr{O}_{2}})$ by

$(U_{1}\times_{\gamma}U_{2})(g):=\sum_{r_{1}s_{1}=r_{2}s_{2}=\gamma}\left(e_{r_{1},s_{1}}\otimes
e_{r_{2},s_{2}}\right)\otimes
u^{(1)}_{r_{1},s_{1}}\bigl{(}\alpha_{r_{2}}(g)\bigr{)}\otimes
u^{(2)}_{r_{2},s_{2}}(g).$

One checks that $U_{1}\times_{\gamma}U_{2}$ is a unitary representation of
$G_{\gamma}$, called the _tensor product of $U_{1}$ and $U_{2}$ twisted by
$\gamma$_. We then have the following theorem which includes the fusion rules
of $\Gamma\bowtie G$ as a special case.

######  Theorem 2.4 (Fusion rules).

Using the above notations, the following hold.

  1. (i)

If $\mathscr{O}_{3}\cap\mathscr{O}_{1}\mathscr{O}_{2}=\emptyset$, then
$\dim\operatorname{Mor}(W_{3},W_{1}\times W_{2})=0$;

  2. (ii)

otherwise $\mathscr{O}_{3}\subseteq\mathscr{O}_{1}\mathscr{O}_{2}$, then

$\displaystyle\dim\operatorname{Mor}(W_{3},W_{1}\times
W_{2})=\dim\operatorname{Mor}_{G_{\gamma}}\left(u^{(3)}_{\gamma,\gamma}|_{G_{\gamma}},U_{1}\times_{\gamma}U_{2}\right)$
$\displaystyle=\dim\operatorname{Mor}_{G_{\mu}}\left(u^{(3)}_{\mu,\mu}|_{G_{\mu}},U_{1}\times_{\mu}U_{2}\right)$

for all $\gamma,\mu\in\mathscr{O}_{3}$.

######  Remark 2.5.

As mentioned at the beginning of § 2.2.2, Theorem 2.4 simplifies our previous
result [MR4345210, Theorem 3.2].

####  2.2.3 Permanence of ${\rm(RD)}$ and polynomial growth

Intuitively speaking, the (dual of) the bicrossed product $\Gamma\bowtie G$
has ${\rm(RD)}$ (resp. polynomial growth) if and only both $\Gamma$ and
$\widehat{G}$ have the same property in a compatible manner. Here the precise
formulation of this compatibility requires the notion of matched pair of
length functions, which arise naturally from the representation theory of
$\Gamma\bowtie G$ as described in § 2.2.2. Again, the following results are
first obtained in [MR4345210] (see also [phdthesis, Chapter I] for a more
polished treatment).

######  Definition 2.6.

Let $l_{\Gamma}$ be a length function on $\Gamma$, $l_{\widehat{G}}$ a length
function on $\widehat{G}$. The pair $\left\lparen\mskip 1.0mul_{\Gamma}\mskip
2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$ is said to be matched, if
there are a family of maps

$\left\\{l_{\mathscr{O}}:\operatorname{Irr}_{\mathscr{O}}(G)\to\mathbb{R}_{\geq
0}\;\middle|\;\mathscr{O}\in\operatorname{Orb}_{\beta}\right\\}$

indexed by $\operatorname{Orb}_{\beta}$, such that the following conditions
are satisfied:

  * •

$l_{\left\\{e_{\Gamma}\right\\}}([\varepsilon_{G}])=0$;

  * •

for all $\mathscr{O}\in\operatorname{Orb}_{\beta}$ and
$[U]\in\operatorname{Irr}_{\mathscr{O}}(G)$, we have
$l_{\mathscr{O}}([U])=l_{\mathscr{O}^{-1}}([U^{{\dagger}}])$;

  * •

for $i=1,2,3$, let $\mathscr{O}_{i}\in\operatorname{Orb}_{\beta}$, and
$[U_{i}]\in\operatorname{Irr}_{\mathscr{O}_{i}}(G)$, with
$U_{i}=\sum_{r,s\in\mathscr{O}_{i}}e_{r,s}\otimes u^{(i)}_{r,s}$ being an
$\mathscr{O}_{i}$-irreducible $\mathscr{O}_{i}$-representation of $G$ on
$\ell^{2}(\mathscr{O}_{i})\otimes\mathscr{H}_{i}$, if

$\dim\operatorname{Mor}_{G_{\gamma}}\left(u^{(3)}_{\gamma,\gamma}|_{G_{\gamma}},U_{1}\times_{\gamma}U_{2}\right)\neq
0$

for some (hence for all, by Theorem 2.4) $\gamma\in\mathscr{O}_{3}$, then

$l_{\mathscr{O}_{3}}([U_{3}])\leq
l_{\mathscr{O}_{1}}([U_{1}])+l_{\mathscr{O}_{2}}([U_{2}]);$

  * •

for all
$[U]\in\operatorname{Irr}_{\left\\{e_{\Gamma}\right\\}}(G)=\operatorname{Irr}(G)$,
we have $l_{\widehat{G}}([U])=l_{\left\\{e_{\Gamma}\right\\}}([U])$ ;

  * •

for all $\mathscr{O}\in\operatorname{Orb}_{\beta}$, the image
$l_{\Gamma}(\mathscr{O})$ is the singleton
$l_{\mathscr{O}}([\varepsilon_{\mathscr{O}}])$, where
$\varepsilon_{\mathscr{O}}:=\sum_{r,s\in\mathscr{O}}e_{r,s}\otimes v_{r,s}$ is
the trivial $\mathscr{O}$-representation (in particular, $l_{\Gamma}$ is
$\beta$-invariant).

If this is the case, we say that the family
$\left\\{l_{\mathscr{O}}\;\middle|\;\mathscr{O}\in\operatorname{Orb}_{\beta}\right\\}$
is affording for the matched pair $\left\lparen\mskip 1.0mul_{\Gamma}\mskip
2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$.

######  Theorem 2.7 (Permanence of polynomial growth).

The following are equivalent:

  1. (i)

$\widehat{\Gamma\bowtie G}$ has polynomial growth;

  2. (ii)

there exists a matched pair of length functions $\left\lparen\mskip
1.0mul_{\Gamma}\mskip 2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$, such
that both $\left\lparen\mskip 1.0mu\widehat{G}\mskip 2.0mu,\mskip
3.0mul_{\widehat{G}}\,\right\rparen$ and $\left\lparen\mskip 1.0mu\Gamma\mskip
2.0mu,\mskip 3.0mul_{\Gamma}\,\right\rparen$ have polynomial growth.

######  Theorem 2.8 (Permanence of ${\rm(RD)}$).

The following are equivalent:

  1. (i)

$\widehat{\Gamma\bowtie G}$ has ${\rm(RD)}$;

  2. (ii)

there exists a matched pair of length functions $\left\lparen\mskip
1.0mul_{\Gamma}\mskip 2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$, such
that $\left\lparen\mskip 1.0mu\widehat{G}\mskip 2.0mu,\mskip
3.0mul_{\widehat{G}}\,\right\rparen$ has polynomial growth and
$\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0mul_{\Gamma}\,\right\rparen$ has ${\rm(RD)}$;

  3. (iii)

there exists a matched pair of length functions $\left\lparen\mskip
1.0mul_{\Gamma}\mskip 2.0mu,\mskip 3.0mul_{\widehat{G}}\,\right\rparen$, such
that both $\left\lparen\mskip 1.0mu\widehat{G}\mskip 2.0mu,\mskip
3.0mul_{\widehat{G}}\,\right\rparen$ and $\left\lparen\mskip 1.0mu\Gamma\mskip
2.0mu,\mskip 3.0mul_{\Gamma}\,\right\rparen$ have ${\rm(RD)}$.

######  Remark 2.9.

When trying to construct concrete examples, the theoretical characterizations
given in Theorems 2.7 and 2.8 have a serious drawback: it is in general very
hard to check that the length functions on $\Gamma$ and $\widehat{G}$ that one
wants to use are actually matched (to determine the existence of an affording
family is quite tricky).

###  2.3 Representation theory of some semi-direct products

For our purposes of constructing length functions on the dual, we need a good
understanding of representations of semi-direct products of the form
$G\rtimes\Lambda$, where $G$ is a compact (meaning quasi-compact and
Hausdorff) group and $\Lambda$ a finite group acting on $G$ by topological
automorphisms. The more general results in [wang2019representations] apply
here by taking the compact quantum group $\mathbb{G}$ studied there to be the
classical compact group $G$. In this case, the irreducible representations can
already be described by Mackey’s method [MR0031489], [MR0098328]. But even in
this special case, the result on fusion rules that we will present seems
fairly recent to the best of our knowledge. We refer our reader to
[wang2019representations] for more details and complete proofs. We will also
freely use the Peter-Weyl theory of projective representations of finite
groups (see e.g. [MR3299063]).

Throughout the rest of § 2.3, we fix a compact group $G$, a finite group
$\Lambda$, and a group homomorphism $\alpha:\Lambda\to\operatorname{Aut}(G)$,
seen as $\Lambda$ acting on $G$ via topological automorphisms, and use
$G\rtimes\Lambda$ to denote the semi-direct with respect to this action. As a
topological space, $G\rtimes\Lambda$ is the Cartesian product $G\times\Lambda$
where $\Lambda$ is equipped with the discrete topology. The multiplication on
$G\rtimes\Lambda$ is given by $(g,r)(h,s)=(g\alpha_{r}(h),rs)$, which makes
$G\rtimes\Lambda$ a compact group.

####  2.3.1 Classification of irreducible representations

The action $\alpha:\Lambda\curvearrowright G$ by topological automorphisms
induces an action
$\widetilde{\alpha}:\Lambda\curvearrowright\operatorname{Irr}(G)$ via
$\lambda\cdot[u]:=[u\circ\alpha_{\lambda}^{-1}]$, which comes from the action
of $\Lambda$ on the (proper) class of all representations of $G$ by
$\lambda\cdot u:=u\circ\alpha_{\lambda}^{-1}$. For each
$x\in\operatorname{Irr}(G)$, one can associate a (unitary) projective
representation $V$ of the isotropy subgroup $\Lambda_{x}$ fixing $x$, as
follows. Take any representation $u\in x$ on some Hilbert space $H$. By
definition, $\lambda\cdot u\simeq u$ if and only if $\lambda\in\Lambda_{x}$.
Hence for any $\lambda\in\Lambda_{x}$, there exists a unitary operator,
uniquely determined up to a constant in the circle group
$\mathbb{T}=\left\\{z\in\mathbb{C}\;\middle|\;\left\lvert
z\right\rvert=1\right\\}$, such that
$V(\lambda)\in\operatorname{Mor}_{G}(\lambda\cdot u,u)$. As
$\operatorname{Mor}_{G}(\lambda\cdot
u,u)=\operatorname{Mor}_{G}(\mu\lambda\cdot u,\mu\cdot u)$ for all
$\lambda,\mu\in\Lambda_{x}$, one checks that
$V(\lambda\mu){V(\mu)}^{\ast}{V(\lambda)}^{\ast}\in\operatorname{Mor}(u,u)$,
hence is a multiple of $\operatorname{id}_{H}$ by a scalar in $\mathbb{T}$.
Thus $V:\Lambda_{x}\to B(H)$, $\lambda\mapsto V(\lambda)$ is a projective
representation of $\Lambda_{x}$ on $H$. The projective representation $V$ is
uniquely determined by the class $x\in\operatorname{Irr}(G)$ in the following
sense: if $V^{\prime}$ is another projective representation of $\Lambda_{x}$
such that $V^{\prime}(\lambda)\in\operatorname{Mor}(\lambda\cdot v,v)$ for
some $v\in x$ and all $\lambda\in\Lambda_{x}$, then for any unitary
equivalence $U\in\operatorname{Mor}(u,v)$, there exists a unique map
$\mathsf{b}:\Lambda_{x}\to\mathbb{T}$, such that for all
$\lambda\in\Lambda_{x}$, we have
$V(\lambda)=\mathsf{b}(\lambda)U^{\ast}V^{\prime}(\lambda)U$. The cohomology
class $[\omega]$ in $H^{2}(\Lambda_{x},\mathbb{T})$ (the second group
cohomology of $\Lambda_{x}$ with coefficients in $\mathbb{T}$ as a trivial
$\Lambda_{x}$-module) of the cocycle $\omega$ of the projective representation
$V$ depends only on $x\in\operatorname{Irr}(G)$. Obviously, we can replace
$\Lambda_{x}$ by any of its subgroup in the above discussion.

######  Notation 2.10.

By $\mathcal{G}_{\mathrm{iso}}(\Lambda)$, we mean the set of finite
intersections of the isotropy subgroups $\Lambda_{x}$,
$x\in\operatorname{Irr}(G)$.

######  Definition 2.11.

Take any $\Lambda_{0}\in\mathcal{G}_{\mathrm{iso}}$, a generalized
representation parameter (abbreviated as GRP later) associated with
$\Lambda_{0}$ (for the semidirect product $G\rtimes\Lambda$) is a triple
$(u,V,v)$, where $u$ is a representation of $G$ on some finite dimensional
Hilbert space $H$, $V$ is a projective representation of $\Lambda_{0}$ on the
same $H$ with $V(\lambda)\in\operatorname{Mor}_{G}(\lambda\cdot u,u)$ for each
$\lambda\in\Lambda_{0}$, and $v$ is a projective representation of
$\Lambda_{0}$ on a possibly different finite dimensional Hilbert space $K$,
such that the cocycle of $v$ is the opposite of that of $V$. The GRP $(u,V,v)$
is called a representation parameter (abbreviated as RP later) if $u$ is
irreducible; and if in addition, $\Lambda_{0}=\Lambda_{[u]}$, we say that
$(u,V,v)$ is a distinguished representation parameter (abbreviated as DRP in
the following).

Given a RP $(u,V,v)$ associated with $\Lambda_{0}$ as in the definition above,
one can associate an irreducible representation
$\mathscr{R}_{\Lambda_{0}}(u,V,v)$ of $G\rtimes\Lambda_{0}$ (which is a
subgroup of $G\rtimes\Lambda$ in which we are interested), called the
representation of $G\rtimes\Lambda_{0}$ parameterized by $(u,V,v)$, as
follows: the carrier space of this representation is $H\otimes K$, and as a
map from $G\rtimes\Lambda_{0}$ to $\mathcal{U}(H\otimes K)$, the
representation $\mathscr{R}_{\Lambda_{0}}(u,V,v)$ sends $(g,\lambda)\in
G\rtimes\Lambda_{0}$ to $u(g)V(\lambda)\otimes v(\lambda)$. Now the induced
representation
$\mathscr{R}(u,V,v):=\operatorname{Ind}_{G\rtimes\Lambda_{0}}^{G\rtimes\Lambda}\bigl{(}\mathscr{R}_{\Lambda_{0}}(u,V,v)\bigr{)}$
is called the representation of $G\rtimes\Lambda$ parameterized by $(u,V,v)$.
If $(u,V,v)$ is distinguished, i.e. if $(u,V,v)$ is a DRP, then
$\mathscr{R}(u,V,v)$ is irreducible.

Fix a $\Lambda_{0}\in\mathcal{G}_{\mathrm{iso}}$, we say two DRP associated
with $\Lambda_{0}$, namely $(u_{1},V_{1},v_{1})$ and $(u_{2},V_{2},v_{2})$,
are equivalent, if $\mathscr{R}_{\Lambda_{0}}(u_{1},V_{1},v_{1})$ and
$\mathscr{R}_{\Lambda_{0}}(u_{2},V_{2},v_{2})$ are equivalent. This
equivalence relation can be characterized more concretely as the equivalence
of the following conditions:

  1. (a)

$(u_{1},V_{1},v_{1})$ and $(u_{2},V_{2},v_{2})$ are equivalent DRPs;

  2. (b)

there exists a mapping $\mathsf{b}:\Lambda_{0}\to\mathbb{T}$ such that the
projective representations $\mathsf{b}V_{1}$ and $V_{2}$ share the same
cocycle, and both
$\operatorname{Mor}_{G}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2})$
and $\operatorname{Mor}_{\Lambda_{0}}(v_{1},v_{2})$ are non-zero;

  3. (c)

there exists a mapping $\mathsf{b}:\Lambda_{0}\to\mathbb{T}$ such that the
projective representations $\mathsf{b}V_{1}$ and $V_{2}$ share the same
cocycle, and both
$\operatorname{Mor}_{G}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2})$
and $\operatorname{Mor}_{\Lambda_{0}}(v_{1},v_{2})$ contain unitary operators.

######  Notation 2.12.

For each $\Lambda_{0}\in\mathcal{G}_{\mathrm{iso}}$, we denote the set of
equivalence classes of DRPs associated with $\Lambda_{0}$ by
$\mathfrak{D}_{\Lambda_{0}}$, and we denote by $\mathfrak{D}$ the union of
$\mathfrak{D}_{\Lambda_{x}}$, $x\in\operatorname{Irr}(G)$.

Given a GRP (resp. RP, resp. DRP) $\mathscr{D}:=(u,V,v)$, the componentwise
contragredient $(u^{c},V^{c},v^{c})$ is still a GRP (resp. RP, resp. DRP),
called the contragredient of $\mathscr{D}$, and is denoted by
$\mathscr{D}^{c}$. Moreover, for each $r\in\Lambda$, denote by
$\operatorname{Ad}_{r}$ the inner automorphisms $x\mapsto rxr^{-1}$ of
$\Lambda$, then the triple

$r\cdot\mathscr{D}:=(r\cdot u,r\cdot V,r\cdot
v)=(u\circ\alpha_{r}^{-1},V\circ\operatorname{Ad}_{r}^{-1}|_{r\Lambda_{0}r^{-1}},v\circ\operatorname{Ad}_{r}^{-1}|_{r\Lambda_{0}r^{-1}})$

is a GRP (resp. RP, resp. DRP) associated with the subgroup
$r\Lambda_{0}r^{-1}\in\mathcal{G}_{\mathrm{iso}}$. One checks easily that
$r\cdot[\mathscr{D}]:=[r\cdot\mathscr{D}]$ gives an action of $\Lambda$ on
$\mathscr{D}$, and this action preserves contragredients.

######  Theorem 2.13 (Classification of irreducible representation of
$G\rtimes\Lambda$).

The mapping

$\displaystyle\Psi:\mathfrak{D}$
$\displaystyle\to\operatorname{Irr}(G\rtimes\Lambda)$
$\displaystyle\mathscr{D}:=[(u,V,v)]\in\mathfrak{D}_{\Lambda_{0}}$
$\displaystyle\mapsto\Psi_{\Lambda_{0}}(\mathscr{D}):=[\mathscr{R}(u,V,v)]$

is a well-defined surjection whose fibers are exactly the orbits of the action
$\Lambda\curvearrowright\mathfrak{D}$. Moreover, $\Psi_{\Lambda_{0}}$, hence
in particular $\Psi$, preserves contragredients.

####  2.3.2 The fusion rules

We give a description of the fusion rules for $G\rtimes\Lambda$ with the help
of a reduction process.

For $i=1,2,3$, consider an irreducible representation $W_{i}$ of
$G\rtimes\Lambda$ parameterized by some DRP
$\mathscr{D}_{i}:=(u_{i},V_{i},v_{i})$ associated with the isotropy subgroup
$\Lambda_{i}:=\Lambda_{[u_{i}]}$. By Theorem 2.13, one only needs to find a
more or less explicit formula for

$\dim\operatorname{Mor}_{G\rtimes\Lambda}(W_{1},W_{2}\times W_{3})$

in order to obtain a description of the desired fusion rules.

We now describe the reduction procedure mentioned above. One checks easily
that $(u_{2}\times u_{3},V_{2}\times V_{3},v_{2}\times v_{3})$ is a GRP
associated with the subgroup
$\Lambda_{0}:=\cap_{i=1}^{3}\Lambda_{i}\in\mathcal{G}_{\mathrm{iso}}$. To fix
notation, let $H_{i}$ (resp. $K_{i}$) be the carrier space of $u_{i}$ (resp.
$v_{i}$). Then there exists a unique subrepresentation $u_{p}$ of $u_{2}\times
u_{3}$ determined by a unique subspace $H_{p}$ of $H_{1}\otimes H_{2}$, such
that $u_{p}$ is maximal among the subrepresentations of $u_{2}\times u_{3}$
that are equivalent to multiples of $u_{1}$, and we denote by $n$ the
multiplicity $\dim\operatorname{Mor}_{G}(u_{1},u_{p})$ of $u_{1}$ in $u_{p}$
(which of course could be $0$).

######  Lemma 2.14.

The following hold.

  1. (1)

The subspace $H_{p}$ is invariant under $V$, and determines a projective
subrepresentation $V_{p}$ of $V$, such that
$V_{p}(\lambda)\in\operatorname{Mor}(\lambda\cdot u_{p},u_{p})$ for all
$\lambda\in\Lambda_{0}$.

  2. (2)

There exists a unique projective representation $V^{\prime}_{p}$ of
$\mathbb{C}^{n}$, such that $V_{p}$ is equivalent to $V_{1}\times
V^{\prime}_{p}$, with the projective representations $V^{\prime}_{p}\times
v_{2}\times v_{3}$ and $v_{1}$ of $\Lambda_{0}$ sharing the same cocycle.

######  Definition 2.15.

We call $\dim\operatorname{Mor}_{\Lambda_{0}}(v_{1},V^{\prime}_{p}\times
v_{2}\times v_{3})$ the incidence number of the triple
$(\mathscr{D}_{1},\mathscr{D}_{2},\mathscr{D}_{3})$ of DRPs.

######  Notation 2.16.

For all $r_{i}\in\Lambda$, $i=1,2,3$, we denote by $m(r_{1},r_{2},r_{3})$ the
incidence number of $(\mathscr{D}_{1},\mathscr{D}_{2},\mathscr{D}_{3})$. It is
easy to check that $m(r_{1},r_{2},r_{3})=m(s_{1},s_{2},s_{3})$ if
$r_{i}\Lambda_{i}=s_{i}\Lambda_{i}$, i.e. the incidence numbers
$m(z_{1},z_{2},z_{3}):=m(r_{1},r_{2},r_{3})$ with $r_{i}$ in the left coset
$z_{i}\in\Lambda/\Lambda_{i}$ is well-defined, i.e. independent of the choice
of the representative $r_{i}\in z_{i}$, $i=1,2,3$. Moreover, denote by
$\Lambda(z_{1},z_{2},z_{3})$ the intersection
$\cap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1}$, which is again independent of the
choices $r_{i}\in z_{i}$, $i=1,2,3$.

######  Theorem 2.17.

The fusion rules for $G\rtimes\Lambda$ is given by the following formula

$\dim\operatorname{Mor}(W_{1},W_{2}\times
W_{3})=\sum_{z_{1}\in\Lambda/\Lambda_{1}}\sum_{z_{2}\in\Lambda/\Lambda_{2}}\sum_{z_{3}\in\Lambda/\Lambda_{3}}\frac{m(z_{1},z_{2},z_{3})}{[\Lambda:\Lambda(z_{1},z_{2},z_{3})]},$

where $[\Lambda:\Lambda_{0}]$ denotes the index of the subgroup $\Lambda_{0}$
of $\Lambda$.

##  3 Nontrivial bicrossed product from semidirect product

Let $G$ be a compact group, $\Gamma$ a discrete group acting on $G$ via
topological automorphisms given by a group morphism
$\tau:\Gamma\to\operatorname{Aut}(G)$. Using these data, one can form the
semidirect product $G\rtimes_{\tau}\Gamma$, which is a locally compact group
whose underlying topological space is the topological product $G\times\Gamma$,
and whose group law is given by

$(g_{1},\gamma_{1})(g_{2},\gamma_{2})=\left(g_{1}\tau_{\gamma_{1}}(g_{2}),\gamma_{1}\gamma_{2}\right).$
(3.1)

Note that the insertion $\iota_{\Gamma}:\Gamma\to G\rtimes_{\tau}\Gamma$,
$\gamma\mapsto(e_{G},\gamma)$ ($e_{G}$ denotes the identity in $G$) is a group
morphism. In particular, the mapping

$\begin{split}\theta:\Gamma&\to\operatorname{Aut}\left(G\rtimes_{\tau}\Gamma\right)\\\
\gamma&\mapsto(\operatorname{Ad}\circ\iota_{\Gamma})(\gamma)=\operatorname{Ad}_{(e_{G},\gamma)}\end{split}$
(3.2)

is a group morphism. For all $(g,r)\in G\rtimes_{\tau}\Gamma$ and
$\gamma\in\Gamma$, we have

$(e_{G},\gamma)(g,r){(e_{G},\gamma)}^{-1}=\left(\tau_{\gamma}(g),\gamma
r\right)\left(e_{G},\gamma^{-1}\right)=\left(\tau_{\gamma}(g),\gamma
r\gamma^{-1}\right).$ (3.3)

Thus as a map from the set $G\times\Gamma$ to itself, we have

$\theta_{\gamma}:=\theta(\gamma)=\tau_{\gamma}\times\operatorname{Ad}_{\gamma}:G\times\Gamma\to
G\times\Gamma.$ (3.4)

Now consider any _finite_ subgroup $\Lambda$ of $\Gamma$. The group morphism
$\theta$ defined in (3.2) restricts to the subgroup $\Lambda$ to give an
action $\Lambda\curvearrowright G\rtimes_{\tau}\Gamma$ by topological
automorphisms. This allows us to form yet another semidirect product
$(G\rtimes_{a}\Gamma)\rtimes_{\theta}\Lambda$, whose underlying topological
space is $G\times\Gamma\times\Lambda$. It is clear that the group law on
$(G\rtimes_{\tau}\Gamma)\rtimes_{\theta}\Lambda$ is given by

$\begin{split}&\hskip
11.99998pt(g_{1},\gamma_{1},r_{1})(g_{2},\gamma_{2},r_{2})=\bigl{(}(g_{1},\gamma_{1})\theta_{r_{1}}(g_{2},\gamma_{2}),r_{1}r_{2}\bigr{)}\\\
&=\Bigl{(}(g_{1},\gamma_{1})\bigl{(}\tau_{r_{1}}(g_{2}),r_{1}\gamma_{2}r_{1}^{-1}\bigr{)},r_{1}r_{2}\Bigr{)}\\\
&=\bigl{(}g_{1}\tau_{\gamma_{1}r_{1}}(g_{2}),\gamma_{1}r_{1}\gamma_{2}r_{1}^{-1},r_{1}r_{2}\bigr{)}.\end{split}$
(3.5)

By (3.5), both the mapping

$\begin{split}\iota_{1,3}:G\rtimes_{\tau}\Lambda&\to(G\rtimes_{\tau}\Gamma)\rtimes_{\theta}\Lambda\\\
(g,r)&\mapsto(g,e,r),\end{split}$ (3.6)

and

$\begin{split}\iota_{2}:\Gamma&\to(G\rtimes_{\tau}\Gamma)\rtimes_{\theta}\Lambda\\\
\gamma&\mapsto(e_{G},\gamma,e)\end{split}$ (3.7)

are injective group morphisms, such that for all $\gamma\in\Gamma$, $(g,r)\in
G\rtimes_{\tau}\Lambda$, we have

$\forall g\in
G,\,\gamma,r\in\Gamma,\quad\iota_{1,3}(g,r)\iota_{2}(r^{-1}\gamma
r)=(g,e,r)(e_{G},r^{-1}\gamma r,e)=(g,\gamma,r),$ (3.8)

which implies that

$\iota_{1,3}(G\rtimes_{\tau}\Lambda)\iota_{2}(\Gamma)=(G\rtimes_{\tau}\Gamma)\rtimes_{\theta}\Lambda.$
(3.9)

It is clear that

$\iota_{1,3}(G\rtimes_{\tau}\Lambda)\cap\iota_{2}(\Gamma)=\left\\{(e_{G},e,e)\right\\}.$
(3.10)

Moreover,

$\begin{split}\forall g\in G,\,\gamma,r\in\Gamma,&\hskip
11.99998pt\iota_{2}(\gamma)\iota_{1,3}(g,r)=(e_{G},\gamma,e)(g,e,r)\\\
&=\bigl{(}\tau_{\gamma}(g),\gamma,r\bigr{)}=\bigl{(}\tau_{\gamma}(g),e,r\bigr{)}(e_{G},r^{-1}\gamma
r,e)\\\ &=\iota_{1,3}(g,r)\iota_{2}(r^{-1}\gamma r).\end{split}$ (3.11)

######  Proposition 3.1.

Let $\Gamma$ be a discrete group, $G$ a compact group, and
$\tau:\Gamma\to\operatorname{Aut}(G)$ a left action of $\Gamma$ on $G$ by
topological automorphisms. If $\Lambda$ is a _finite_ subgroup of $\Gamma$,
then $\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0muG\rtimes_{\tau}\Lambda\,\right\rparen$ is a matched pair of groups with
left action

$\begin{split}\alpha^{\Lambda}:\Gamma\times(G\rtimes_{\tau}\Lambda)&\to
G\rtimes_{\tau}\Lambda\\\
\bigl{(}\gamma,(g,r)\bigr{)}&\mapsto\bigl{(}\tau_{\gamma}(g),r\bigr{)},\end{split}$
(3.12)

and right action

$\begin{split}\beta^{\Lambda}:\Gamma\times(G\rtimes_{\tau}\Lambda)&\to\Gamma\\\
\bigl{(}\gamma,(g,r)\bigr{)}&\mapsto r^{-1}\gamma r.\end{split}$ (3.13)

Moreover, the following hold.

  1. (1)

The action $\alpha^{\Lambda}$ is trivial if and only if $\tau$ is trivial;

  2. (2)

The action $\beta^{\Lambda}$ is trivial if and only if $\Lambda\subseteq
Z(\Gamma)$, where $Z(\Gamma)$ is the centre of $\Gamma$.

###### Proof.

That $\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0muG\rtimes_{\tau}\Lambda\,\right\rparen$ is a matched pair with the actions
$\alpha^{\Lambda}$ and $\beta^{\Lambda}$ follows from (3.9), (3.10) and
(3.11). (1) and (2) are direct consequences of the definition of
$\alpha^{\Lambda}$ and $\beta^{\Lambda}$. ∎

##  4 More notation

For the convenience of our discussion, we now introduce and fix some notation
related to the bicrossed product of the matched pair $\left\lparen\mskip
1.0mu\Gamma\mskip 2.0mu,\mskip 3.0muG\rtimes_{\tau}\Lambda\,\right\rparen$
with the actions $\alpha^{\Lambda}$ and $\beta^{\Lambda}$, as described in §
3.

The bicrossed product of the matched pair $\left\lparen\mskip
1.0mu\Gamma\mskip 2.0mu,\mskip 3.0muG\rtimes_{\tau}\Lambda\,\right\rparen$ is
denoted by
$\Gamma{}_{\alpha^{\Lambda}}\bowtie_{\beta^{\Lambda}}(G\rtimes_{\tau}\Lambda)$.
When there is no risk of confusion, we often omit the actions and simply write
$G\rtimes\Lambda$ and $\Gamma\bowtie(G\rtimes\Lambda)$. Moreover,
$\operatorname{Aut}(G)$ denotes the group of topological automorphisms of $G$.

The isotropy subgroup of $G\rtimes\Lambda$ fixing $\gamma\in\Gamma$ with
respect to the action $\beta^{\Lambda}$ is easily seen to be
$G\rtimes\Lambda_{\gamma}$, where

$\Lambda_{\gamma}:=\left\\{r\in\Lambda\;\middle|\;\gamma r=r\gamma\right\\},$
(4.1)

i.e. $\Lambda_{\gamma}$ is the centralizer of $\gamma$ in $\Lambda$.

For $x\in\operatorname{Irr}(G)$, $\gamma\in\Gamma$, we denote the isotropy
subgroup of $\Lambda_{\gamma}$ fixing $x$ with respect to the action
$\Lambda_{\gamma}\curvearrowright\operatorname{Irr}(G)$,
$(r,[u])\mapsto[u\circ\tau_{r}]$ by $\Lambda_{\gamma,x}$, i.e.

$\Lambda_{\gamma,x}:=\left\\{r\in\Lambda_{\gamma}\;\middle|\;r\cdot
x=x\right\\}.$ (4.2)

We also need to fix some notation concerning the representation theory of
$\Gamma\bowtie(G\rtimes\Lambda)$, this is where the preliminaries described in
§ 2 come into play.

Let $\gamma\in\Gamma$. Suppose $\Lambda_{0}$ is an isotropy subgroup of
$\Lambda_{\gamma}$ with respect to the action
$\Lambda_{\gamma}\curvearrowright\operatorname{Irr}(G)$. Let
$\mathfrak{D}_{\gamma,\Lambda_{0}}$ denote the set of equivalent DRPs
(distinguished representation parameters) (see § 2.3.1) associated with
$\Lambda_{0}$, and let

$\Psi_{\gamma,\Lambda_{0}}:\mathfrak{D}_{\gamma,\Lambda_{0}}\to\operatorname{Irr}(G\rtimes\Lambda_{\gamma})$
(4.3)

be the injection used to classify irreducible unitary representations of
$G\rtimes\Lambda_{\gamma}$ as in Theorem 2.13. Let $\mathfrak{D}_{\gamma}$ be
the set of equivalency classes of all DRPs for the semidirect product
$G\rtimes\Lambda_{\gamma}$, we then have an action of $\Lambda_{\gamma}$ on
the class of all DRPs for $G\rtimes\Lambda_{\gamma}$, which passes to the
quotient and yields an action
$\Lambda_{\gamma}\curvearrowright\mathfrak{D}_{\gamma}$ as described in §
2.3.1. We thus have the classification _surjection_

$\begin{split}\Psi_{\gamma}:\mathfrak{D}_{\gamma}&\to\operatorname{Irr}(G\rtimes\Lambda_{\gamma})\\\
[(u,V,v)]\in\mathfrak{D}_{\gamma,\Lambda_{0}}&\mapsto\Psi_{\gamma,\Lambda_{0}}\bigl{(}[(u,V,v)]\bigr{)},\end{split}$
(4.4)

whose fibers are exactly the orbits for the action
$\Lambda_{\gamma}\curvearrowright\mathfrak{D}_{\gamma}$.

When $\gamma=e_{\Gamma}$, we then have $\Lambda_{\gamma}=\Lambda$ and we write
$\Psi_{e_{\Gamma}}$ simply as $\Psi$.

We use $\operatorname{Orb}_{\beta^{\Lambda}}$ to denote the set of
$\beta^{\Lambda}$-orbits. For each
$\mathscr{O}\in\operatorname{Orb}_{\beta^{\Lambda}}$, let
$\mathfrak{R}_{\mathscr{O}}$ be the mapping from the class of
$\mathscr{O}$-representations of $G\rtimes\Lambda$ to the class of finite
dimensional unitary representations of the bicrossed product
$\Gamma\bowtie(G\rtimes\Lambda)$ as in § 2.2.2, and let
$\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)$ denote the set of
equivalency classes of $\mathscr{O}$-irreducible
$\mathscr{O}$-representations. We thus have the classification bijection

$\begin{split}\mathfrak{R}:\coprod_{\mathscr{O}\in\operatorname{Orb}_{\beta^{\Lambda}}}\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)&\to\operatorname{Irr}\bigl{(}\Gamma\bowtie(G\rtimes\Lambda)\bigr{)}\\\
[U]\in\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)&\mapsto[\mathfrak{R}_{\mathscr{O}}(U)].\end{split}$
(4.5)

##  5 Proof of Theorem 1.1

The purpose of this section is to establish Theorem 1.1 which gives a
sufficient condition for the (dual of) bicrossed product
$\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$ to have polynomial growth (resp.
property ${\rm(RD)}$).

We first establish the following technical result.

######  Lemma 5.1.

Suppose $l_{\widehat{G}}:\operatorname{Irr}(G)\to\mathbb{R}_{\geq 0}$ is a
$\Gamma$-invariant length function on $\widehat{G}$, i.e.
$l_{\widehat{G}}([u^{x}\circ\tau_{\gamma}])=l_{\widehat{G}}(x)$ whenever
$\gamma\in\Gamma$, $u^{x}\in x\in\operatorname{Irr}(G)$, and $l_{\Gamma}$ is a
$\beta^{\Lambda}$-invariant length function on $\Gamma$. Then

$\displaystyle
l_{\widehat{G\rtimes\Lambda}}:\operatorname{Irr}(G\rtimes\Lambda)$
$\displaystyle\to\mathbb{R}_{\geq 0}$ $\displaystyle\Psi([(u,V,v)])$
$\displaystyle\mapsto l_{\widehat{G}}([u])$

is a well-defined length function on $\widehat{G\rtimes\Lambda}$ such that the
pair $\left\lparen\mskip 1.0mul_{\Gamma}\mskip 2.0mu,\mskip
3.0mul_{\widehat{G\rtimes\Lambda}}\,\right\rparen$ is matched.

###### Proof.

The fact that $l_{\widehat{G\rtimes\Lambda}}$ is well-defined (i.e. does not
depend on the choice of the DRP $(u,V,v)$) follows from Theorem 2.13 and the
$\Lambda$-invariance of $l_{\widehat{G}}$. We now show that the pair
$\left\lparen\mskip 1.0mul_{\Gamma}\mskip 2.0mu,\mskip
3.0mul_{\widehat{G\rtimes\Lambda}}\,\right\rparen$ of length functions is
matched.

For all $\mathscr{O}\in\operatorname{Orb}_{\beta^{\Lambda}}$, define
$l_{\mathscr{O}}:\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)\to\mathbb{R}_{\geq
0}$ via the following procedure. Take any $\gamma\in\mathscr{O}$, and let
$\varPhi_{\gamma}:\operatorname{Irr}(G\rtimes\Lambda_{\gamma})\to\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)$
be the canonical bijection as in Proposition 2.2. To avoid over-complication
of our notation, we often implicitly identify
$\operatorname{Irr}(G\rtimes\Lambda_{\gamma})$ with
$\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)$ via the bijection
$\varPhi_{\gamma}$, when doing so won’t cause a risk of confusion. For each
DRP $(u,V,v)$ of $G\rtimes\Lambda_{\gamma}$, let

$l_{\mathscr{O}}\left(\Psi_{\gamma}\bigl{(}[(u,V,v)]\bigr{)}\right):=l_{\widehat{G}}([u])+l_{\Gamma}(\gamma).$
(5.1)

By Theorem 2.13 again, we see that (5.1) yields a well-defined mapping
$l_{\mathscr{O}}:\operatorname{Irr}_{\mathscr{O}}(G\rtimes\Lambda)\to\mathbb{R}_{\geq
0}$. It is clear that $l_{\left\\{e_{\Gamma}\right\\}}=l_{\widehat{G}}$ via
the identification of
$\operatorname{Irr}_{\left\\{e_{\Gamma}\right\\}}(G\rtimes\Lambda)$ with
$\operatorname{Irr}(G\rtimes\Lambda)$ by $\varPhi_{e_{\Gamma}}$. Moreover, we
have
$[\varepsilon_{\mathscr{O}}]=\Psi_{\gamma}([\varepsilon_{G},\varepsilon_{\Lambda_{\gamma}},\varepsilon_{\Lambda_{\gamma}}])$,
so that

$l_{\Gamma}(\gamma)=l_{\widehat{G}}([\varepsilon_{G}])+l_{\Gamma}(\gamma)=l_{\mathscr{O}}([\varepsilon_{\mathscr{O}}]).$

Therefore, to finish the proof, it remains to show that
${(l_{\mathscr{O}})}_{\mathscr{O}\in\operatorname{Orb}_{\beta^{\Lambda}}}$ is
an affording family in the sense of Definition 2.6. By definition, it is clear
that

$l_{\left\\{e_{\Gamma}\right\\}}([\varepsilon_{G\rtimes\Lambda}])=l_{\widehat{G}}([\varepsilon_{G}])+l_{\Gamma}(e_{\Gamma})=0.$

The condition $l_{\mathscr{O}}([U])=l_{\mathscr{O}^{-1}}([U^{{\dagger}}])$ can
also be easily checked. Indeed, if $[U]$ is given by
$\Psi_{\gamma}\bigl{(}[(u,V,v)]\bigr{)}$, then $[U^{{\dagger}}]$ is given by
$\Psi_{\gamma^{-1}}\bigl{(}[(\overline{u},V^{c},v^{c})]\bigr{)}$ (see Theorem
2.3). By (5.1), we now have

$l_{\mathscr{O}}([U])=l_{\widehat{G}}([u])+l_{\Gamma}(\gamma)=l_{\widehat{G}}([\overline{u}])+l_{\Gamma}(\gamma^{-1})=l_{\mathscr{O}^{-1}}([U^{{\dagger}}]).$
(5.2)

By Definition 2.6, it remains only to establish the following claim.

Claim. For $i=1,2,3$, let
$\mathscr{O}_{i}\in\operatorname{Orb}_{\beta^{\Lambda}}$, and
$[U_{i}]\in\operatorname{Irr}_{\mathscr{O}_{i}}(G\rtimes\Lambda)$, with

$U_{i}=\sum_{r,s\in\mathscr{O}_{i}}e_{r,s}\otimes u^{(i)}_{r,s}$

being an $\mathscr{O}_{i}$-irreducible $\mathscr{O}_{i}$-representation of
$G\rtimes\Lambda$ on $\ell^{2}(\mathscr{O}_{i})\otimes\mathscr{H}_{i}$. If

$\dim\operatorname{Mor}_{G\rtimes\Lambda_{\gamma}}\left(u^{(3)}_{\gamma,\gamma}|_{G\rtimes\Lambda_{\gamma}},U_{1}\times_{\gamma}U_{2}\right)\neq
0$ (5.3)

for some (hence for all) $\gamma\in\mathscr{O}_{3}$, then

$l_{\mathscr{O}_{3}}([U_{3}])\leq
l_{\mathscr{O}_{1}}([U_{1}])+l_{\mathscr{O}_{2}}([U_{2}]).$ (5.4)

Before proving the claim, we remark that until now, only the
$\Lambda$-invariance of $l_{\widehat{G}}$ is needed. The hypothesis that
$l_{\widehat{G}}$ is $\Gamma$-invariant will play an important role in the
proof of the claim as we will presently see.

We now prove the claim. Suppose $[U_{i}]$ is given by some
$\Psi_{\gamma_{i}}\bigl{(}[(u_{i},V_{i},v_{i})]\bigr{)}$ and let
$\Lambda_{i}:=\Lambda_{\gamma_{i},[u_{i}]}$ for each $i=1,2,3$. Define
$\mu\cdot u:=u\circ\tau_{\mu}$ to be the left action of $\Gamma$ on the class
of finite dimensional unitary representation of $G$, and let $M(u)$ denote the
vector space of matrix coefficients of $u$. Using the character formulae for
$U_{1}\times_{\gamma}U_{2}$ and for
$\Psi_{\gamma_{3}}\bigl{(}[(u_{3},V_{3},v_{3})]\bigr{)}$, as well as the
construction of $\Psi_{\gamma_{3}}$, we see that as elements in
$\operatorname{Pol}(G)\otimes C(\Lambda_{\gamma})$, we have

$\chi\left(u_{\gamma,\gamma}^{(3)}|_{G\rtimes\Lambda_{\gamma}}\right)\in\operatorname{Vect}\left(\bigcup_{r\in\Lambda}M(r\cdot
u_{3})\right)\otimes C(\Lambda_{\gamma})\subseteq\operatorname{Pol}(G)\otimes
C(\Lambda_{\gamma}),$ (5.5)

and

$\chi\left(U_{1}\times_{\gamma}U_{2}\right)\in\operatorname{Vect}\Bigl{(}\bigl{[}\Gamma\cdot
M(u_{1})\bigr{]}\bigl{[}\Gamma\cdot M(u_{2})\bigr{]}\Bigr{)}\otimes
C(\Lambda_{\gamma}),$ (5.6)

where

$\forall i=1,2,\qquad\Gamma\cdot
M(u_{i}):=\bigcup_{r_{i}\in\Gamma}M(r_{i}\cdot u_{i})$

and $\bigl{[}\Gamma\cdot M(u_{1})\bigr{]}\bigl{[}\Gamma\cdot M(u_{2})\bigr{]}$
denotes product of form $\varphi_{1}\varphi_{2}\in\operatorname{Pol}(G)$ where
$\varphi_{i}\in\Gamma\cdot M(u_{i})$ for $i=1,2$. By (5.5), (5.6) and a simple
calculation using the Haar state on $C(G)\otimes
C(\Lambda_{\gamma})=C(G\rtimes\Lambda_{\gamma})$, it is clear that (5.3)
implies the existence of $r\in\Lambda$, $r_{1},r_{2}\in\Gamma$, such that
$M(r\cdot u_{3})$ and $M(r_{1}\cdot u_{1})\cdot M(r_{2}\cdot u_{2})$ are _not_
orthogonal with respect to the Haar measure on $G$. Since the representation
$r\cdot u_{3}$ of $G$ is irreducible, this forces that

$\dim\operatorname{Mor}_{G}\bigl{(}r\cdot u_{3},(r_{1}\cdot
u_{1})\times(r_{2}\cdot u_{2})\bigr{)}\neq 0.$

Hence

$r\cdot[u_{3}]\subseteq(r_{1}\cdot[u_{1}])\otimes(r_{2}\cdot[u_{2}]).$ (5.7)

Since $l_{\widehat{G}}$ is a $\Gamma$-invariant length function, by (5.7), we
have

$l_{\widehat{G}}([u_{3}])=l_{\widehat{G}}(r\cdot[u_{3}])\leq
l_{\widehat{G}}(r_{1}\cdot[u_{1}])+l_{\widehat{G}}(r_{2}\cdot[u_{2}])=l_{\widehat{G}}([u_{1}])+l_{\widehat{G}}([u_{2}]).$
(5.8)

On the other hand, (5.3) also implies that
$\gamma_{3}\in\mathscr{O}_{3}\subseteq\mathscr{O}_{1}\mathscr{O}_{2}$, so
there is $s_{i}\in\mathscr{O}_{i}$, $i=1,2$, such that
$s_{1}s_{2}=\gamma_{3}$. Using the fact that $l_{\Gamma}$ is a
$\beta^{\Lambda}$-invariant length function, we have

$l_{\Gamma}(\gamma_{3})=l_{\Gamma}(s_{1}s_{2})\leq
l_{\Gamma}(s_{1})+l_{\Gamma}(s_{2})=l_{\Gamma}(\gamma_{1})+l_{\Gamma}(\gamma_{2}).$
(5.9)

By (5.8), (5.9) and (5.1) again, we have

$\displaystyle l_{\mathscr{O}_{3}}([U_{3}])$
$\displaystyle=l_{\widehat{G}}([u_{3}])+l_{\Gamma}(\gamma_{3})$
$\displaystyle\leq
l_{\widehat{G}}([u_{1}])+l_{\widehat{G}}([u_{2}])+l_{\Gamma}(\gamma_{1})+l_{\Gamma}(\gamma_{2})$
$\displaystyle=l_{\mathscr{O}_{1}}([U_{1}])+l_{\mathscr{O}_{2}}([U_{2}]).$

This finishes the proof of the claim, and hence the lemma. ∎

Now Theorem 1.1 follows from Lemma 5.1, Theorem 2.7 and Theorem 2.8.

######  Remark 5.2.

It is however, unknown to the author that whether the polynomial growth (resp.
${\rm(RD)}$) of the dual $\widehat{\Gamma\bowtie(G\rtimes\Lambda)}$ implies
the existence of a _$\Gamma$ -invariant_ length function $l_{\widehat{G}}$ on
$\widehat{G}$ witnessing the polynomial growth (resp. ${\rm(RD)}$) of
$\widehat{G}$. Later we will show that if the composition
$\Gamma\xrightarrow{\tau}\operatorname{Aut}(G)\to\operatorname{Out}(G)$ has
finite image, then the converse of Theorem 1.1 also holds (Theorem 1.2).

##  6 Invariance of length functions and proof of Theorem 1.2

In this section, we partially treat the difficulty of the technical assumption
on the $\Gamma$-invariance of the length function $l_{\widehat{G}}$ on
$\widehat{\Gamma}$ that witnesses the polynomial growth or $(RD)$ of
$\widehat{G}$, as presented in Theorem 1.1. The results here will be used in §
7 in which we give some concrete examples of bicrossed products whose duals
have $(RD)$ but do _not_ have polynomial growth. We also point out that the
examples that will be given in § 8 do _not_ fit into this framework, thus we
only have a partial understanding of the situation. In particular, the tools
developed here lead to a natural proof of Theorem 1.2.

We begin by considering a technical lemma on the Fourier transform and the
Sobolev-$0$-norm in the context of compact quantum groups of Kac type.

######  Lemma 6.1.

Let $\mathbb{H}$ be a compact quantum group of Kac type. Suppose
$\theta:C(\mathbb{H})\to C(\mathbb{H})$ is an automorphism of
$C^{\ast}$-algebras that intertwines the comultiplication $\Delta$ of
$\mathbb{H}$ ($\theta$ is an automorphism of the quantum group $\mathbb{H}$).
Then there exists an automorphism $\widehat{\theta}$ of the involutive algebra
$c_{c}(\widehat{\mathbb{H}})$, such that

$\forall a\in
c_{c}(\widehat{\mathbb{H}}),\quad\mathcal{F}_{\mathbb{H}}\left(\widehat{\theta}(a)\right)=\theta\bigl{(}\mathcal{F}_{\mathbb{H}}(a)\bigr{)}\quad\text{
and
}\quad\left\|\widehat{\theta}(a)\right\|_{\mathbb{H},0}=\left\|a\right\|_{\mathbb{H},0}.$
(6.1)

###### Proof.

Choose a complete set of representatives
$\left\\{u^{x}\;\middle|\;x\in\operatorname{Irr}(\mathbb{H})\right\\}$ for
$\operatorname{Irr}(\mathbb{H})$, and denote the finite dimensional Hilbert
space underlying the unitary representation $u^{x}$ by $\mathscr{H}_{x}$, so
that

$c_{c}\bigl{(}\widehat{\mathbb{H}}\bigr{)}=\bigoplus_{x\in\operatorname{Irr}(\mathbb{H})}^{\operatorname{alg}}B\bigl{(}\mathscr{H}_{x}\bigr{)}.$

For each finite dimensional unitary representation $u\in
B(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{H})$ of $\mathbb{H}$ on
$\mathscr{H}$, since $\theta$ is an automorphism of $\mathbb{H}$, the unitary
operator

$\theta_{\ast}(u):=(\operatorname{id}\otimes\theta)(u)\in
B(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{H})$ (6.2)

remains a unitary representation of $\mathbb{H}$ on the same space
$\mathscr{H}$. It is clear that $\theta_{\ast}$ also passes to a bijection of
the set $\operatorname{Irr}(\mathbb{H})$ to itself, which we still denote by
$\theta_{\ast}$ by abuse of notation, via
$\theta_{\ast}([u])=[\theta_{\ast}(u)]$. In particular, for each
$x\in\operatorname{Irr}(\mathbb{H})$, we have
$\left[u^{\theta_{\ast}(x)}\right]=\theta_{\ast}(x)=\left[\theta_{\ast}(u^{x})\right]$,
thus there exists a _unitary_

$T_{x}\in\operatorname{Mor}_{\mathbb{H}}\left(u^{\theta_{\ast}(x)},\theta_{\ast}(u^{x})\right)\subseteq
B\left(\mathscr{H}_{\theta_{\ast}(x)},\mathscr{H}_{x}\right),$

which is uniquely determined up to a multiple of a scalar in $\mathbb{T}$.

Take any

$a={(a_{x})}_{x\in\operatorname{Irr}(\mathbb{H})}=\sum_{x\in\operatorname{Irr}(\mathbb{H})}a_{x}\in
c_{c}(\widehat{\mathbb{H}}),$ (6.3)

where the sum is finite (meaning all but finitely many $a_{x}\in
B(\mathscr{H}_{x})$ is $0$). For each $x\in\operatorname{Irr}(\mathbb{H})$, we
set

$b_{\theta_{\ast}(x)}:=T_{x}^{\ast}a_{x}T_{x}\in
B\left(\mathscr{H}_{\theta_{\ast}(x)}\right).$ (6.4)

Then

$\dim\bigl{(}\theta_{\ast}(x)\bigr{)}=\dim x.$ (6.5)

By the choice of $T_{x}$, we have

$\begin{split}&\hskip
11.99998pt\left(\operatorname{Tr}_{\mathscr{H}_{\theta_{\ast}(x)}}\otimes\operatorname{id}\right)\Bigl{(}u^{\theta_{\ast}(x)}(b_{\theta_{\ast}(x)}\otimes
1)\Bigr{)}\\\
&=\left(\operatorname{Tr}_{\mathscr{H}_{\theta_{\ast}(x)}}\otimes\operatorname{id}\right)\Bigl{(}(T_{x}^{\ast}\otimes
1)\bigl{[}[\theta_{\ast}(u^{x})](a_{x}\otimes 1)\bigr{]}(T_{x}\otimes
1)\Bigr{)}\\\
&=\left(\operatorname{Tr}_{\mathscr{H}_{x}}\otimes\operatorname{id}\right)\Bigl{(}[\theta_{\ast}(u^{x})](a_{x}\otimes
1)\Bigr{)}\\\
&=\theta\left[\left(\operatorname{Tr}_{\mathscr{H}_{x}}\otimes\operatorname{id}\right)\Bigl{(}[u^{x}(a_{x}\otimes
1)\Bigr{)}\right],\end{split}$ (6.6)

and

$\begin{split}&\hskip
11.99998pt\left(\operatorname{Tr}_{\mathscr{H}_{\theta_{\ast}(x)}}\otimes\operatorname{id}\right)\bigl{(}b_{\theta_{\ast}(x)}^{\ast}b_{\theta_{\ast}(x)}\bigr{)}\\\
&=\left(\operatorname{Tr}_{\mathscr{H}_{\theta_{\ast}(x)}}\otimes\operatorname{id}\right)\bigl{(}T_{x}^{\ast}a_{x}^{\ast}a_{x}T_{x}\bigr{)}\\\
&=\left(\operatorname{Tr}_{\mathscr{H}_{\theta_{\ast}(x)}}\otimes\operatorname{id}\right)(a_{x}^{\ast}a_{x}).\end{split}$
(6.7)

We now define

$\widehat{\theta}(a):=\sum_{x\in\operatorname{Irr}(\mathbb{H})}b_{\theta_{\ast}(x)}$
(6.8)

Since
$\theta_{\ast}:\operatorname{Irr}(\mathbb{H})\to\operatorname{Irr}(\mathbb{H})$
is a bijection, it is clear that (6.8) defines an automorphism
$\widehat{\theta}$ of the involutive algebra $c_{c}(\widehat{\mathbb{H}})$.
Finally, (6.1) follows from (6.6) and (6.7). ∎

######  Proposition 6.2.

Let $\mathbb{H}=\bigl{(}C(\mathbb{H}),\Delta\bigr{)}$ be a compact quantum
group of Kac type. Suppose $\Theta$ is a _finite_ subgroup of
$\operatorname{Aut}\bigl{(}C(\mathbb{H}),\Delta\bigr{)}$. The following are
equivalent.

  1. (1)

There exists a length function $l$ on $\widehat{\mathbb{H}}$ and
$P(X)\in\mathbb{R}[X]$, such that

$\forall k\in\mathbb{N},\quad a\in
Q_{l,k}c_{c}(\widehat{\mathbb{H}})\implies\left\|\mathcal{F}_{\mathbb{H}}(a)\right\|\leq
P(k)\left\|a\right\|_{\mathbb{H},0},$ (6.9)

where
$Q_{l,n}:=\sum_{x\in\operatorname{Irr}(\mathbb{H}),\,l(x)<n+1}p_{x}\in\ell^{\infty}(\widehat{\mathbb{H}})$.

  2. (2)

There exists a _$\Theta$ -invariant_ length function $l_{\Theta}$ on
$\widehat{\mathbb{H}}$ and $Q(X)\in\mathbb{R}[X]$, such that

$\forall k\in\mathbb{N},\quad a\in
Q_{l_{\Theta},k}c_{c}(\widehat{\mathbb{H}})\implies\left\|\mathcal{F}_{\mathbb{H}}(a)\right\|\leq
Q(k)\left\|a\right\|_{\mathbb{H},0}.$ (6.10)

###### Proof.

Obviously (2) implies (1).

Now suppose (1) holds and let’s prove (2). Let
$n=\left\lvert\Theta\right\rvert$ and suppose $\theta_{1},\ldots,\theta_{n}$
form an enumeration of all elements of $\Theta$. Let $l_{i}$ denote the length
function $l\circ{(\theta_{i})}_{\ast}$ on $\widehat{\mathbb{H}}$ (see the
discussion after (6.2) in the proof of Lemma 6.1). Put

$l_{\Theta}:={1\over\left\lvert\Theta\right\rvert}\sum_{i=1}^{n}l_{i},$ (6.11)

then it is clear that $l_{\Theta}$ is a $\Theta$-invariant length function on
$\widehat{\mathbb{H}}$. For each $k\in\mathbb{N}$, define

$F_{\Theta,k}:=\left\\{x\in\operatorname{Irr}(\mathbb{H})\;\middle|\;l_{\Theta}(x)<k+1\right\\},$
(6.12)

and for $i=1,\ldots,n$, put

$F_{i,k}:=\left\\{x\in\operatorname{Irr}(\mathbb{H})\;\middle|\;l_{\theta_{i}}(x)<k+1\right\\}.$
(6.13)

By (6.9), (6.13) and (6.11), we have

$F_{\Theta,k}\subseteq\bigcup_{i=1}^{n}F_{i,k}.$ (6.14)

Define

$\begin{split}\xi:F_{\Theta,k}&\to\left\\{1,\ldots,n\right\\}\\\
x&\mapsto\inf\left\\{i\;\middle|\;x\in F_{i,k}\right\\}.\end{split}$ (6.15)

Note that (6.14) guarantees that $\xi$ is well-defined.

We now prove (6.10) holds for some suitable polynomial $Q(X)\in\mathbb{R}[X]$,
which will finish the proof. Since $a\in
Q_{l_{\Theta},k}c_{c}(\widehat{\mathbb{H}})$, there exists a finite subset $F$
of $F_{\Theta,k}$, such that

$a=\sum_{x\in F}a_{x}=\sum_{i=1}^{n}a_{i},$ (6.16)

where for each $i$,

$a_{i}:=\sum_{x\in F\cap\xi^{-1}(i)}a_{x}\in Q_{l_{i},k}.$ (6.17)

By Lemma 6.1 and (1), we have

$\forall i=1,\ldots,n,\quad\left\|\mathcal{F}_{\mathbb{H}}(a_{i})\right\|\leq
P(k)\left\|a_{i}\right\|_{\mathbb{H},0},$ (6.18)

hence

$\begin{split}\left\|\mathcal{F}_{\mathbb{H}}(a)\right\|^{2}&\leq{\left(\sum_{i=1}^{n}\left\|\mathcal{F}_{\mathbb{H}}(a_{i})\right\|\right)}^{2}\leq
n\left(\sum_{i=1}^{n}\left\|\mathcal{F}_{\mathbb{H}}(a_{i})\right\|^{2}\right)\\\
&\leq
n{[P(k)]}^{2}\left(\sum_{i=1}^{n}\left\|a_{i}\right\|_{\mathbb{H},0}^{2}\right)=\left\lvert\Theta\right\rvert{[P(k)]}^{2}\left\|a\right\|_{\mathbb{H},0}^{2}.\end{split}$
(6.19)

Thus taking $Q(X)=\sqrt{\left\lvert\Theta\right\rvert}P(X)\in\mathbb{R}[X]$,
we have (6.10). ∎

######  Corollary 6.3.

The following are equivalent:

  1. (1)

$\Gamma$ has polynomial growth (resp. $(RD)$);

  2. (2)

there exists a $\beta^{\Lambda}$-invariant length function $l_{\Gamma}$ on
$\Gamma$, such that $\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0mul_{\Gamma}\,\right\rparen$ has polynomial growth (resp. $(RD)$).

###### Proof.

This follows from Proposition 6.2 by letting
$\Theta=\left\\{\operatorname{Ad}_{r}\in\operatorname{Aut}(\Gamma)\;\middle|\;r\in\Lambda\right\\}$
and $\mathbb{H}=\widehat{\Gamma}$. ∎

###### Proof of Theorem 1.2.

We begin by observing more closely the action
$\Gamma\curvearrowright\operatorname{Irr}(G)$. It is clear that this action is
actually given by $\operatorname{Aut}(G)$ acting on $G$, composed with the
group morphism $\tau:\Gamma\to\operatorname{Aut}(G)$ with respect to which we
form the semidirect product (see the beginning of § 3). More precisely, there
is a natural action
$\operatorname{Aut}(G)\curvearrowright\operatorname{Irr}(G)$ by letting
$(\theta,[u])\mapsto[\theta_{\ast}(u)]$, and the action
$\Gamma\curvearrowright\operatorname{Irr}(G)$ is given by
$(\gamma,x)\mapsto\tau(\gamma)\cdot x$. By definition, one has

$\operatorname{Inn}(G)\subseteq\bigcap_{x\in\operatorname{Irr}(G)}{[\operatorname{Aut}(G)]}_{x},$
(6.20)

where

${[\operatorname{Aut}(G)]}_{x}:=\left\\{\theta\in\operatorname{Aut}(G)\;\middle|\;\theta\cdot
x=x\right\\}.$

Thus passing to the quotient, it is in fact
$\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$ that acts
on $\operatorname{Irr}(G)$ (inner automorphisms fixes equivalence classes of
representations by definition). Thus to talk about the $\Gamma$ invariance of
a given length function $l$ on $\widehat{G}$, it suffices to consider the
invariance of $l$ under the image of the composition of group morphisms
$\tau\colon\Gamma\to\operatorname{Aut}(G)$ and the canonical projection
$\operatorname{Aut}(G)\to\operatorname{Out}(G)$.

With the above consideration in mind, Theorem 1.2 now follows from Theorem
1.1, Proposition 6.2 (posing $\mathbb{H}=G$ and
$\Theta=\operatorname{Image}(\widetilde{\tau})$) and Corollary 6.3. ∎

##  7 Examples of bicrossed products with rapid decay but not polynomial
growth–part I

In this section, we construct some examples of bicrossed products whose duals
have property ${\rm(RD)}$ but do not have polynomial growth, using the
framework developed in § 6. We shall frequently use Jolissaint’s theorem on
rapid decay of amalgamated product of groups, which we record here for
convenience of the reader.

######  Theorem 7.1 (Jolissaint).

Suppose $\Gamma_{1}$, $\Gamma_{2}$ are two discrete groups with property
$(RD)$, $A$ is a finite group, $j_{i}:A\hookrightarrow\Gamma_{i}$ is an
injective group morphism for $i=1,2$, then the amalgamated product
$\Gamma_{1}\ast_{A}\Gamma_{2}$ with respect to $j_{1}$, $j_{2}$ also has
property $(RD)$.

###### Proof.

This is part of [MR943303, Theorem 2.2.2]. ∎

We will refer Theorem 7.1 as Jolissaint’s theorem hereafter.

######  Example 7.2.

Take
$\Gamma=\operatorname{PSL}_{2}(\mathbb{Z})\simeq(\mathbb{Z}/2\mathbb{Z})\ast(\mathbb{Z}/3\mathbb{Z})$,
with the isomorphism determined by identifying $\mathbb{Z}/2\mathbb{Z}$ with
the cyclic group generated by $s\in\Gamma$, and $\mathbb{Z}/3\mathbb{Z}$ with
the cyclic group generated by $t\in\Gamma$, where

$s=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\quad\text{ and }\quad
t=\begin{pmatrix}0&-1\\\ 1&1\end{pmatrix}.$

(see e.g. [MR2391387, Example E.10 on page 476]). Let $G$ be any compact
connected real Lie group that admits an element $x\in G$ of order $2$, and an
element $y$ of order $3$, such that $\left\\{x,y\right\\}\not\subseteq Z(G)$
(e.g. $G=\operatorname{SO}(3,\mathbb{R})$, $x$ is any rotation by $\pi$, $y$
any rotation by $2\pi/3$), where $Z(G)$ is the center of $G$. Now the mapping
$s\mapsto\operatorname{Ad}_{x}$, $t\mapsto\operatorname{Ad}_{y}$ determines a
unique group morphism

$\tau:\Gamma\to\operatorname{Inn}(G)\subseteq\operatorname{Aut}(G)$

so $\widetilde{\tau}:\Gamma\to\operatorname{Out}(G)$ is trivial (hence of
finite image). Put $\Lambda<\Gamma$ to be $\langle s\rangle$ or $\langle
t\rangle$ (the subgroup generated by $s$ or $t$ respectively). Since
$\Lambda\not\subseteq Z(\Gamma)$, it follows from the choice of $x$ and $y$
that the resulted bicrossed product
$\mathbb{G}:=\Gamma{}_{\alpha^{\Lambda}}\bowtie_{\beta^{\Lambda}}(G\rtimes_{\tau}\Lambda)$
is nontrivial (Proposition 3.1).

By Jolissaint’s theorem, $\operatorname{PSL}_{2}(\mathbb{Z})$ has ${\rm(RD)}$,
but $\operatorname{PSL}_{2}(\mathbb{Z})$ does not have polynomial growth since
it is not virtually nilpotent (by Gromov’s famous theorem on polynomial
growth, see [MR623534]), and by Vergnioux [MR2329000], $\widehat{G}$ has
polynomial growth, thus Theorem 1.2 applies and we see that
$\widehat{\mathbb{G}}$ has ${\rm(RD)}$ but not polynomial growth.

######  Example 7.3.

Let $G$ be any compact group with $\widehat{G}$ having polynomial growth (e.g.
all connected compact real Lie group), and $\Lambda$ a finite subgroup of
$\operatorname{Aut}(G)$. Take $\Gamma$ to be a nontrivial semidirect product
of the free group $\mathbb{F}_{2}$ on two generators (here $\mathbb{F}_{2}$
can be replaced by any discrete group with $(RD)$ but without polynomial
growth) with $\Lambda$ (in particular, $\Lambda$ is nontrivial). Then the
obvious action of $\Lambda$ on $G$ and the canonical projection
$\mathbb{F}_{2}\rtimes\Lambda\to\Lambda$ together yield a nontrivial left
action $\tau$ of $\Gamma$ on $G$ by topological automorphisms. The same
reasoning as in Example 7.2 shows that $\Gamma\bowtie(G\rtimes\Lambda)$ is
also a bicrossed product whose dual has ${\rm(RD)}$ but not polynomial growth.

Using Theorem 1.2, many more examples can be constructed in the same spirit.

##  8 Examples of bicrossed products with rapid decay but not polynomial
growth–part II

Despite of the fact that Theorem 1.2 yields many interesting concrete examples
of bicrossed products with property ${\rm(RD)}$ as shown in § 7, it is worth
pointing out that the restriction the finiteness of the image
$\operatorname{Image}(\widetilde{\tau})$ is too strong to include many other
interesting examples, which we will now show in this section. To make the
contrast even more dramatic, we show how to construct examples of nontrivial
bicrossed product of the form $\Gamma\bowtie(G\rtimes\Lambda)$ whose dual has
${\rm(RD)}$ but _not_ polynomial growth, while
$\operatorname{Image}(\widetilde{\tau})$ as in Theorem 1.2 is infinite (hence
Theorem 1.2 no longer applies).

We begin with a simple result in finite group theory.

######  Lemma 8.1.

If $A$ is a finite abelian group, then there exists infinitely many non-
isomorphic finite abelian groups $B$, such that $A$ is isomorphic to a
subgroup of $\operatorname{Aut}(B)$.

###### Proof.

Since $A$ is a direct sum of finite cyclic groups, without loss of generality,
we may assume $A$ is cyclic of order $n$, with $a$ as a generator. Set $B$ to
be the $n$-fold direct sum of any nontrivial finite abelian group $C$, and
define $\sigma(a)\in\operatorname{Aut}(B)$ to be the permutation

$(c_{1},\ldots,c_{n})\mapsto(c_{2},\ldots,c_{n},c_{1}).$

Then it is clear that

$\displaystyle\sigma:A$ $\displaystyle\to\operatorname{Aut}(B)$ $\displaystyle
a^{m}$ $\displaystyle\mapsto{[\sigma(a)]}^{m}$

is a well-defined injective group morphism. ∎

As we will see later, Theorem 1.2 no longer applies for the examples
constructed below in this section due to the violation of the hypothesis of
the finiteness of $\operatorname{Image}(\widetilde{\tau})$. We will thus have
to resort to Theorem 1.1 to prove the ${\rm(RD)}$ of the dual of the bicrossed
product $\Gamma\bowtie(G\rtimes\Lambda)$. Here, the
$\beta^{\Lambda}$-invariance of the length function on $\Gamma$ is not a
problem thanks to Corollary 6.3. But the $\Gamma$-invariance of the length
function on $\widehat{G}$ requires some work.

######  Lemma 8.2.

Let $\Xi_{1},\Xi_{2},\ldots$ be a sequence of finite discrete (hence compact)
groups. The product group $\prod_{i=1}^{\infty}\operatorname{Aut}(\Xi_{i})$
naturally acts pointwise on the direct sum $\oplus_{i=1}^{\infty}\Xi_{i}$,
giving the canonical inclusion
$\prod_{i=1}^{\infty}\operatorname{Aut}(\Xi_{i})\subseteq\operatorname{Aut}(\oplus_{i=1}^{\infty}\Xi_{i})$.
Under this setting, there exists a
$\prod_{i=1}^{\infty}\operatorname{Aut}(\Xi_{i})$-invariant length function
$l$ on the discrete group $\oplus_{i=1}^{\infty}\Xi_{i}$, such that the pair
$\left\lparen\mskip 1.0mu\oplus_{i=1}^{\infty}\Xi_{i}\mskip 2.0mu,\mskip
3.0mul\,\right\rparen$ has polynomial growth.

###### Proof.

Let $N_{i}=\left\lvert\Xi_{i}\right\rvert$ for all $i\in\mathbb{N}_{>0}$ and
take $M_{k}=\prod_{i=1}^{k}N_{i}$ for all $k\in\mathbb{N}$ (we make the
convention that $M_{0}=1$). Let $e_{i}$ be the neutral element of the group
$\Xi_{i}$, and denote the characteristic function of
$\Xi_{i}\setminus\left\\{e_{i}\right\\}$ by $\chi_{i}$. Define

$\displaystyle l:\oplus_{i=1}^{\infty}\Xi_{i}$
$\displaystyle\to\mathbb{R}_{\geq 0}$ $\displaystyle(\xi_{i})$
$\displaystyle\mapsto\sum_{i=1}^{\infty}\chi_{i}(\xi_{i})M_{i}.$

Then it is clear that $l$ is a
$\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$-invariant length function on
$\oplus_{i=1}^{\infty}\Xi_{i}$. Moreover, for all $n\in\mathbb{N}_{>0}$, there
exists a unique $k\geq 1$, such that $M_{k-1}\leq n<M_{k}$. Then, by the
definition of $l$, we have

$\left\\{\xi=(\xi_{i})\in\oplus_{i=1}^{\infty}\Xi_{i}\;\middle|\;l(\xi)<n\right\\}\subseteq\left\\{(\xi_{i})\in\oplus_{i=1}^{\infty}\Xi_{i}\;\middle|\;\forall
i\geq k,\,\xi_{i}=e_{i}\right\\}.$

Thus

$\left\lvert\left\\{\xi=(\xi_{i})\in\oplus_{i=1}^{\infty}\Xi_{i}\;\middle|\;l(\xi)<n\right\\}\right\rvert\leq\prod_{i=1}^{k-1}N_{i}=M_{k-1}\leq
n.$

In particular, $\left\lparen\mskip 1.0mu\oplus_{i=1}^{\infty}\Xi_{i}\mskip
2.0mu,\mskip 3.0mul\,\right\rparen$ has polynomial growth. ∎

We are now prepared to construct new examples of bicrossed product of the form
$\Gamma\bowtie(G\rtimes\Lambda)$ that don’t fit into the framework of Theorem
1.2.

######  Example 8.3.

Let $\Lambda$ be any nontrivial finite abelian group. By Lemma 8.1, one can
take a sequence of finite abelian groups ${(G_{i})}_{i=1}^{\infty}$, such that
$\Lambda$ is isomorphic to a subgroup of $\operatorname{Aut}(G_{i})$ for each
$i=1,2,\ldots$ via an injective group morphism
$j_{i}:\Lambda\hookrightarrow\operatorname{Aut}(G_{i})$. Equip each $G_{i}$
with the discrete topology, and $G:=\prod_{i=1}^{\infty}G_{i}$ the product
topology. Then $G$ is a compact abelian group. In particular, the character
group $\chi(G)$ of $G$, as a set of one-dimensional representations, is a
complete set of representatives of $\operatorname{Irr}(G)$. By Pontryagin’s
duality, we have $\chi(G)\simeq\oplus_{i=1}^{\infty}\chi(G_{i})$, and it is
clear that length functions on $\widehat{G}$ become exactly length functions
on the discrete group $\chi(G)$ of continuous characters of $G$. But as a
finite abelian group, each $G_{i}$ is isomorphic to $\chi(G_{i})$ (albeit the
isomorphism is not natural in the categorical sense). Thus Lemma 8.2 shows
that there exists a $\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$-invariant
length function $l_{G}$ on $\widehat{G}$, such that $\left\lparen\mskip
1.0mu\widehat{G}\mskip 2.0mu,\mskip 3.0mul_{G}\,\right\rparen$ has polynomial
growth, where we’ve used the canonical inclusion
$\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})\subseteq\operatorname{Aut}(G)$.

The construction of $\Gamma$ takes some more work which we now explain. First
we take $\Lambda^{\prime}$ to be any nontrivial finite group and set
$\Gamma_{1}$ to be the free product $\Lambda\ast\Lambda^{\prime}$. It follows
from Jolissaint’s theorem and Gromov’s theorem that $\Gamma_{1}$ has
${\rm(RD)}$ but not polynomial growth. Define
$j:\Lambda\hookrightarrow\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$ to be
the mapping
$\lambda\mapsto\bigl{(}j_{1}(\lambda),j_{2}(\lambda),\cdots\bigr{)}$. Take any
infinite discrete subgroup $\Gamma_{2}^{\prime}$ of
$\oplus_{i=1}^{\infty}\operatorname{Aut}(G_{i})\subseteq\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$
such that $j(\Lambda)$ is contained in the normalizer of $\Gamma_{2}^{\prime}$
in $\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$. Obviously $j(\Lambda)$ and
$\Gamma_{2}^{\prime}$ intersect trivially, thus the subgroup of
$\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$ generated by $j(\Lambda)$ and
$\Gamma_{2}^{\prime}$ is the (internal) semidirect product of
$\Gamma_{2}^{\prime}$ with $j(\Lambda)$, which we denote by $\Gamma_{2}$.
Since $\oplus_{i=1}^{\infty}\operatorname{Aut}(G_{i})$ has polynomial growth
by Lemma 8.2, it follows that $\Gamma_{2}^{\prime}$, hence $\Gamma_{2}$ (note
that $[\Gamma_{2}:\Gamma_{2}^{\prime}]=\left\lvert\Lambda\right\rvert$ is
finite) has polynomial growth. In particular, $\Gamma_{2}$ has ${\rm(RD)}$,
and $j:\Lambda\hookrightarrow\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})$
restricts an injective group morphism, which we still denote by $j$, from
$\Lambda$ into $\Gamma_{2}$. To facilitate our discussion, we identify
$\Lambda$ with its copy in $\Gamma_{1}=\Lambda\ast\Lambda^{\prime}$ and in
$\Gamma$ via $j$. This allows us to form the amalgamated product of
$\Gamma_{1}$ and $\Gamma_{2}$ over $A$, which we denote by $\Gamma$.
Jolissaint’s theorem applies again and proves that $\Gamma$ has ${\rm(RD)}$.
Moreover, $\Gamma$ does not have polynomial growth since its subgroup
$\Gamma_{1}$ does not. We also make the obvious identification of $\Lambda$
with $j(\Lambda)$ in $\Gamma$. By Corollary 6.3, there exists a
$\Lambda$-invariant length function $l_{\Gamma}$ on $\Gamma$, meaning
$l_{\Gamma}=l_{\Gamma}\circ\operatorname{Ad}_{r}$ for all $r\in\Lambda$, such
that $\left\lparen\mskip 1.0mu\Gamma\mskip 2.0mu,\mskip
3.0mul_{\Gamma}\,\right\rparen$ has ${\rm(RD)}$.

Finally, let’s explain how the action, which is a group morphism
$\tau:\Gamma\to\operatorname{Aut}(G)$, is defined. The trivial group morphism
$\Lambda^{\prime}\to\operatorname{Aut}(G)$, together with
$j:\Lambda\to\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})\subseteq\operatorname{Aut}(G)$
and the universal property of free products, yields a group morphism
$\tau_{1}:\Gamma_{1}\to\operatorname{Aut}(G)$. Let $\tau_{2}$ be the simple
inclusion
$\Gamma_{2}\hookrightarrow\prod_{i=1}^{\infty}\operatorname{Aut}(G_{i})\subseteq\operatorname{Aut}(G)$.
It is clear that $\tau_{1}$ and $\tau_{2}$ agree on $\Lambda$, thus the
universal property of the amalgamated product
$\Gamma_{1}\ast_{\Lambda}\Gamma_{2}$ applies and determines a unique group
morphism $\tau:\Gamma\to\operatorname{Aut}(G)$ that restricts to $\tau_{i}$ on
the corresponding copy of $\Gamma_{i}$. We can finally construct the bicrossed
product $\Gamma\bowtie(G\rtimes\Lambda)$, and we conclude by Theorem 1.1 that
the dual of $\Gamma\bowtie(G\rtimes\Lambda)$ has ${\rm(RD)}$ (it does not have
polynomial growth because of Theorem 2.7 and the fact that $\Gamma$ does not
have polynomial growth).

It is clear by our construction that
$\operatorname{Image}(\tau)=\Gamma_{2}\subseteq\operatorname{Aut}(G)=\operatorname{Out}(G)$
is infinite, thus Theorem 1.2 does not apply.

## References