On Bohr compactifications and profinite completions of group extensions

Abstract

Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

1. Introduction

There are two distinguished compact groups associated to a general topological group G. A Bohr compactification (respectively, a profinite completion) of G is a pair consisting of a compact (respectively, profinite) group K and a continuous homomorphism $\beta\,:\, G \to K $ with dense image satisfying the following universal property: for every compact group (respectively, profinite group) L and every continuous homomorphism $\alpha\,:\, G \to L$ , there exists a continuous homomorphism $\alpha'\,:\, K\to L$ such that the diagram

commutes. Bohr compactifications and profinite completions $(K, \beta) $ of G are unique in the following sense: if $(K', \beta')$ is a pair consisting of a compact (respectively, profinite) group K’ and a continuous homomorphism $\beta'\,:\, G \to K'$ with dense image satisfying the same universal property, then there exists an isomorphism $f\,:\, K \to K'$ of topological groups such that $\beta' = f \circ \beta$ . Concerning existence, we give below (Proposition 4) models of Bohr compactifications and profinite completions. For more on Bohr compactifications, see [ Dix77 , section 16], [ BdlH , 4·C] or [ Wei40 , chapter VII]; for more details on profinite completions, see [ RZ00 ].

We will often denote by $({\rm Bohr}(G), \beta_G)$ and $({\rm Prof}(G), \alpha_G)$ a Bohr compactification and a profinite completion of G. In the sequel, for two topological groups H and L, we write $H\cong L$ if H and L are topologically isomorphic.

The universal property of ${\rm Bohr}(G)$ gives rise to a continuous surjective homomorphism $\alpha\,:\, {\rm Bohr}(G) \to {\rm Prof}(G)$ such that $\alpha_G = \alpha \circ \beta_G$ . It is easy to see (see [ Bek23 , proposition 7]) that the kernel of $\alpha$ is ${\rm Bohr}(G)_0$ , the connected component of ${\rm Bohr}(G),$ and so

\begin{equation*}{\rm Prof}(G)\cong {\rm Bohr}(G)/ {\rm Bohr}(G)_0.\end{equation*}

Every continuous homomorphism $G_1 \xrightarrow{f} G_2$ of topological groups induces continuous homomorphisms

\begin{equation*}{\rm Bohr}(G_1) \xrightarrow{{\rm Bohr}(\,f)} {\rm Bohr}(G_2) \qquad \text{and}\qquad{\rm Prof}(G_1) \xrightarrow{{\rm Prof}(\,f)} {\rm Prof}(G_2)\end{equation*}

such that $\beta_{G_2} \circ f= {\rm Bohr}(\,f)\circ\beta_{G_1}$ and $\alpha_{G_2} \circ f= {\rm Prof}(\,f)\circ\alpha_{G_1}.$

Consider the category TGrp of topological groups, with objects the topological groups and morphisms the continuous homomorphisms between topological groups. The Bohr compactification and the profinite completion are covariant functors

\begin{equation*}{\rm Bohr}\,:\, \textbf{TGrp} \to \textbf{CGrp} \qquad \text{and}\qquad {\rm Prof}\,:\,\textbf{TGrp} \to \textbf{PGrp}\end{equation*}

from TGrp to the subcategory CGrp of compact groups and the subcategory PGrp of profinite groups.

Assume that we are given an extension

\begin{equation*}\quad\qquad\qquad\qquad \qquad \qquad 1 \longrightarrow N \mathop{\longrightarrow}\limits^i G \mathop{\longrightarrow}\limits^p G/N \longrightarrow 1\qquad\qquad\quad\qquad\qquad\qquad \, (*) \end{equation*}

of topological groups. The functors ${\rm Bohr}$ and ${\rm Prof}$ are right exact and so the diagrams

\begin{equation*} {\rm Bohr}(N) \xrightarrow{{\rm Bohr}(i)} {\rm Bohr}(G) \xrightarrow{{\rm Bohr}(p)} {\rm Bohr}(G/N) \longrightarrow 1 \end{equation*}

and

\begin{equation*} {\rm Prof}(N) \xrightarrow{{\rm Prof}(i)} {\rm Prof}(G) \xrightarrow{{\rm Prof}(p)} {\rm Prof}(G/N) \longrightarrow 1 \end{equation*}

are exact; this means that

\begin{align} & {\rm Bohr}(p)\ {\rm and}\ {\rm Prof}(p)\ {\rm are\ surjective\ and}\\ & {{\rm Ker}} ({\rm Bohr}(p))=\overline{\beta_G(N)}\ {\rm and}\ {{\rm Ker}} ({\rm Prof}(p))=\overline{\alpha_G(N)}, \end{align}

where $\overline{A}$ denotes the closure of a subset A; these facts are well known and easy to prove (see, e.g., [ HK01 , lemma 2·2] and [ RZ00 , proposition 3·2·5]; see also Proposition 7 below). However, the functors ${\rm Bohr}$ and ${\rm Prof}$ are not left exact, that is, ${\rm Bohr}(i)\,:\, {\rm Bohr}(N) \to {\rm Bohr}(G)$ and ${\rm Prof}(i)\,:\, {\rm Prof}(N) \to {\rm Prof}(G)$ are in general not injective (see e.g. the examples given by Corollaries F and G below).

For now on, we will deal only with locally compact groups. and with split extensions. So, we will consider locally compact groups $G=N\rtimes H$ which are a semi-direct product of a normal closed subgroup N and a closed subgroup H. It is easy to see that ${\rm Bohr}(G)$ , respectively ${\rm Prof}(G)$ , is a semi-direct product of $\overline{\beta_G(N)}$ with $\overline{\beta_G(H)}$ , respectively of $\overline{\alpha_G(N)}$ with $\overline{\alpha_G(H)}$ (see [ Jun78 , GZ11 ]). Our results give a precise description of the structure of these semi-direct products.

Denote by $\widehat{N}_{ \rm fd}$ the set of equivalence classes (modulo unitary equivalence) of irreducible finite dimensional unitary representations of N. Every such representation $\sigma\,:\, N\to U(n)$ gives rise to the unitary representation ${\rm Bohr}({\sigma})\,:\,{\rm Bohr}(N)\to U(n) $ of ${\rm Bohr}(N)$ ; here (and elsewhere) we identify ${\rm Bohr}(U(n))$ with $U(n).$

Observe that H acts on $ \widehat{N}_{\rm fd}$ : for $\sigma\in \widehat{N}_{\rm fd}$ and $h\in H,$ the conjugate representation $\sigma^h \in \widehat{N}_{\rm fd}$ is defined by $\sigma^ h(n)=\sigma( h^{-1} n h)$ for all $n\in N.$

Define $\widehat{N}_{ \rm fd}^{H-{\rm per}}$ as the set of $\sigma\in \widehat{N}_{ \rm fd}$ with finite H-orbit.

Observe that, due to the universal property of ${\rm Bohr}(N)$ , the group H acts by automorphisms on ${\rm Bohr}(N)$ . However, this action does not extend in general to an action of ${\rm Bohr}(H)$ on ${\rm Bohr}(N).$

Our first result shows that ${\rm Bohr}(G)$ is a split extension of ${\rm Bohr}(H)$ by an appropriate quotient of ${\rm Bohr}(N).$

Theorem A. Let $G=N\rtimes H$ be a semi-direct product of locally compact groups. Let $\varphi_N\,:\, {\rm Bohr}(N)\to \overline{\beta_G(N)}$ and $\varphi_H\,:\, {\rm Bohr}(H)\to \overline{\beta_G(H)}$ be the maps such that $\varphi_N\circ \beta_N= \beta_G|_N$ and $\varphi_H\circ \beta_H= \beta_G|_H$ Set

\begin{equation*}C\,:\!=\,\bigcap_{\sigma \in \widehat{N}_{ \rm fd}^{H-{\rm per}}} {\rm {Ker}} ({\rm Bohr}(\sigma)).\end{equation*}

1. (i) We have ${\rm {Ker}} \varphi_N =C$ and so $\varphi_N$ induces a topological isomorphism $\overline{\varphi_N}\,:\, {\rm Bohr}(N)/C \to \overline{\beta_G(N)}.$

2. (ii) $\varphi_H\,:\, {\rm Bohr}(H)\to \overline{\beta_G(H)}$ is a topological isomorphism.

3. (iii) The action of H by automorphisms on ${\rm Bohr}(N)$ induces an action of ${\rm Bohr}(H)$ by automorphisms on ${\rm Bohr}(N)/C$ and the maps $\overline{\varphi_N}$ and $\varphi_H$ give rise to an isomorphism

   \begin{equation*} {\rm Bohr}(G)\cong ({\rm Bohr}(N)/C) \rtimes {\rm Bohr}(H).\end{equation*}

We turn to the description of ${\rm Prof}(G).$ Let $\widehat{N}_{ \rm finite} $ be the set of equivalence classes of irreducible unitary representations $\sigma$ of N with finite image $\sigma(N).$ Observe that the action of H on $ \widehat{N}_{\rm fd}$ preserves $\widehat{N}_{ \rm finite}.$ Let $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ be the subset of $\widehat{N}_{ \rm finite}$ of representations with finite H-orbit. Every $\sigma\in \widehat{N}_{ \rm finite}$ gives rise to the unitary representation ${\rm Prof}({\sigma}) $ of ${\rm Prof}(N)$ .

A result completely similar to Theorem A holds for ${\rm Prof}(G)$ .

Theorem B. Let $G=N\rtimes H$ be a semi-direct product of locally compact groups. Let $\psi_N\,:\, {\rm Prof}(N)\to \overline{\alpha_G(N)}$ and $\psi_H\,:\, {\rm Prof}(H)\to \overline{\alpha_G(H)}$ be the maps such that $\psi_N\circ \alpha_N= \alpha_G|_N$ and $\psi_H\circ \alpha_H= \alpha_G|_H$ Set

\begin{equation*}D\,:\!=\,\bigcap_{\sigma \in \widehat{N}_{ \rm finite}^{H-{\rm per}}} {\rm {Ker}} ({\rm Prof}({\sigma})).\end{equation*}

1. (i) We have ${\rm {Ker}} \psi_N =D$ and so $\psi_N$ induces a topological isomorphism $\overline{\psi_N}\,:\, {\rm Prof}(N)/D \to \overline{\alpha_G(N)}.$

2. (ii) $\psi_H\,:\, {\rm Prof}(H)\to \overline{\alpha_G(H)}$ is a topological isomorphism.

3. (iii) The action of H by automorphisms on ${\rm Prof}(N)$ induces an action of ${\rm Prof}(H)$ by automorphisms on ${\rm Prof}(N)/D$ and the maps $\overline{\psi_N}$ and $\psi_H$ give rise to an isomorphism

   \begin{equation*}{\rm Prof}(G) \cong ({\rm Prof}(N)/D) \rtimes {\rm Prof}(H) .\end{equation*}

When N is a finitely generated (discrete) group, we obtain the following well known result (see [ GZ11 , proposition 2·6]).

Corollary C. Assume that N is finitely generated. Then ${\rm Prof}(G) \cong {\rm Prof}(N) \rtimes {\rm Prof}(H) .$

In the case where N is abelian, we can give a more explicit description of the quotients ${\rm Bohr}(N)/C$ and ${\rm Prof}(N)/D$ appearing in Theorems A and B. Recall that, in this case, the dual group $\widehat{N}$ is the group of continuous homomorphisms from N to the circle group $\textbf{S}^1$ . We will also consider the subgroup $\widehat{N}_{\rm fin}$ of $\chi\in \widehat{N}$ with finite image $\chi(N)$ , that is, with values in the subgroup of m-th roots of unity in $\textbf{C}$ for some integer $m\geq 1.$ Observe also that $\widehat{N}^{H-{\rm per}}$ and $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ are subgroups of $\widehat{N}$ .

Corollary D. Assume that N is an abelian locally compact group. Let $\widehat{N}^{H-{\rm per}}$ and $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ be equipped with the discrete topology. Let A and B be their respective dual groups. Then

\begin{equation*}{\rm Bohr}(G) \cong A \rtimes {\rm Bohr}(H) \qquad\text{and} \qquad {\rm Prof}(G) \cong B \rtimes {\rm Prof}(H) .\end{equation*}

Recall that G is maximally almost periodic, or MAP, if $\widehat{G}_{ \rm fd}$ separates its points (equivalently, if $\beta_G\,:\, G\to {\rm Bohr}(G)$ is injective); recall also that G is residually finite, or RF, if $\widehat{G}_{ \rm finite}$ separates its points (equivalently, if $\alpha_G\,:\, G\to {\rm Prof}(G)$ is injective).

Corollary E. Let $G=N\rtimes H$ be a semi-direct product of locally compact groups.

1. (i) G is MAP if and only if H is MAP and $\widehat{N}_{ \rm fd}^{H-{\rm per}}$ separates the points of N.

2. (ii) G is RF if and only if H is RF and $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ separates the points of N.

We give an application of our results to wreath products. Let $H, \Lambda$ be groups, X a non empty set, and $H \curvearrowright X$ an action of H on X. Then H acts on the direct sum $\oplus_{x\in X} \Lambda,$ by shifting the indices. The (permutational) wreath product, denoted $\Lambda\wr_{X} H,$ is the semidirect product

\begin{equation*}\Lambda\wr_{X} H\,:\!=\, (\!\oplus_{x\in X} \Lambda) \rtimes H.\end{equation*}

When the action of H on X is simply transitive, we obtain the standard wreath product denoted $\Lambda\wr H.$ Observe that $\Lambda^{\rm Ab}\wr_X H$ is a quotient of $\Lambda\wr_X H,$ where $\Lambda^{\rm Ab}$ is the abelianization $\Lambda/[\Lambda, \Lambda]$ of $\Lambda.$

Initially, we formulated the next two corollaries only for standard wreath products; the extension of these results to more general wreath products was suggested to us by the referee.

Corollary F. Let $H, \Lambda$ be groups, and let $H \curvearrowright X$ be a transitive action of H on a set X. Let $\Lambda\wr_{X} H$ be equipped with the discrete topology.

1. (i) When X is finite, we have

   \[\begin{aligned}&{\rm Bohr}(\Lambda\wr_X H)\cong \left( \oplus_{x\in X}{\rm Bohr}(\Lambda)\right) \rtimes {\rm Bohr}(H) \ \text{and}\\&{\rm Prof}(\Lambda\wr_X H)\cong \left( \oplus_{x\in X}{\rm Prof}(\Lambda)\right) \rtimes {\rm Prof}(H).\end{aligned}\]

2. (ii) When X is infinite, the quotient map $\Lambda\wr_X H\to \Lambda^{\rm Ab}\wr_X H$ induces isomorphisms

   \begin{equation*}{\rm Bohr}(\Lambda\wr_X H)\cong {\rm Bohr}( \Lambda^{\rm Ab}\wr_X H) \ \text{and}\ {\rm Prof}(\Lambda\wr_X H)\cong {\rm Prof}( \Lambda^{\rm Ab}\wr_X H)\end{equation*}
   In particular, if $\Lambda$ is perfect (that is, $\Lambda=[\Lambda, \Lambda]$ ), the quotient map $\Lambda\wr_X H\to H$ induces isomorphisms
   \begin{equation*}{\rm Bohr}(\Lambda\wr_X H)\cong {\rm Bohr}(H) \ \text{and}\ {\rm Prof}(\Lambda\wr_X H)\cong {\rm Prof}( H).\end{equation*}

The following definition was suggested to us by the referee.

Definition 1. An action $H \curvearrowright X$ of a group H on a set X is residually finite or RF, if, for any pair $x_1, x_2$ of distinct elements of X, there exists a finite index subgroup L of H such that $L x_1\neq L x_2.$

Observe that $H \curvearrowright X$ is RF if and only if $H \curvearrowright Y$ is RF for every H-orbit $Y\subset X.$ Observe also that, when $H \curvearrowright X$ is simply transitive, the action $H \curvearrowright X$ is RF if and only if the group H is RF.

Item (ii) of the following result was proved, with different methods, in [ Gru57 , theorem 3·2] for standard wreath products and in [ Cor14 , proposition 1·7] for permutational wreath products.

Corollary G. Let $\Lambda, H$ be groups, and let $H \curvearrowright X$ be an action of H on a set X. Let $\Lambda\wr_{X} H$ be equipped with the discrete topology.

Assume that $\Lambda$ has at least two elements.

1. (i) The group $\Lambda\wr_X H$ is MAP if and only if $\Lambda$ and H are MAP, and either

   1. – $\Lambda$ is abelian and $H \curvearrowright X$ is RF, or

   2. – X is finite.

2. (ii) ([ Gru57 ], [ Cor14 ]) The group $\Lambda\wr_X H$ is RF if and only if $\Lambda$ and H are RF, and either

   1. – $\Lambda$ is abelian and $H \curvearrowright X$ is RF, or

   2. – X is finite.

Remark 2.

1. (i) The Bohr compactification of an abelian locally compact group A is easy to describe: ${\rm Bohr}(A)$ can be identified with $\widehat{\Gamma},$ where $\Gamma= \widehat{A}$ is viewed as discrete group; in case A is finitely generated, a more precise description of ${\rm Bohr}(A)$ is available (see [ Bek23 , proposition 11]).

2. (ii) Provided ${\rm Bohr}(H)$ and ${\rm Prof}(H)$ are known, Corollary F together with Corollary D give, in view of (i), a complete description of the Bohr compactification and the profinite completion of any wreath product $\Lambda\wr_X H$ in case X is infinite.

3. (iii) Bohr compactifications of group and semigroup extensions have been studied by several authors, in a more abstract and less explicit setting ([ DL83 , JL81 , Jun78 , JM02 , Lan72 , Mil83 ]); profinite completions of group extensions appear at numerous places in the literature ([ GZ11 , RZ00 ]).

This paper is organised as follows. Section 2 contains some general facts about Bohr compactifications and profinite completions as well as some reminders on projective representations. In Section 3, we give the proof of Theorems A and B. Section 4 contains the proof of the corollaries. Section 5 is devoted to the explicit computation of the Bohr compactification and profinite completions for two groups: the lamplighter group $(\textbf{Z}/n\textbf{Z}) \wr \textbf{Z}$ and the Heisenberg group H(R) over an arbitrary commutative ring R.

2. Preliminaries

2·1. Models for Bohr compactifications and profinite completions

Let G be a topological group. We give well known models for ${\rm Bohr}(G)$ and ${\rm Prof}(G).$ For this, we use finite dimensional unitary representations of G, that is, continuous homomorphisms $\pi\,:\, G\to U(n)$ for some integer $n\geq 1.$ We denote by $\widehat{G}_{\rm fd}$ the set of equivalence classes of irreducible finite dimensional unitary representations of G. Let $\widehat{G}_{\rm finite}$ be the subset of $\widehat{G}_{\rm fd}$ consisting of representations $\pi$ with finite image $\pi(G).$

For a compact (respectively, profinite) group K, the set $\widehat{K}_{\rm fd}$ (respectively, $\widehat{K}_{\rm finite}$ ) coincides with the dual space $\widehat{K}$ , that is, the set of equivalence classes of unitary representations of K.

A useful tool for the identification of ${\rm Bohr}(G)$ or ${\rm Prof}(G)$ is given by the following proposition; for the easy proof, see [ Bek23 , propositions 5 and 6].

Proposition 3

1. (i) Let K be a compact group and $\beta\,:\, G\to K$ a continuous homomorphism with dense image; then $(K, \beta)$ is a Bohr compactification of G if and only if the map $\widehat{\beta}\,:\,\widehat{K}\to \widehat{G}_{\rm fd},$ given by $\widehat{\beta}(\pi)= \pi\circ \beta,$ is surjective.

2. (ii) Let L a be profinite group and $\alpha\,:\, G\to L$ a continuous homomorphism with dense image; then $(L, \alpha)$ is a profinite completion of G if and only if the map $\widehat{\beta}\,:\,\widehat{L}\to \widehat{G}_{\rm finite},$ given by $\widehat{\beta}(\pi)= \pi\circ \beta,$ is surjective.

The following proposition is an immediate consequence of Proposition 3.

Proposition 4. Choose families

\begin{equation*}(\pi_i\,:\, G\to U(n_i))_{i\in I} \quad \text{and} \quad (\sigma_j\,:\, G\to U(n_j))_{j\in J}\end{equation*}

of representatives for the sets $\widehat{G}_{\rm fd}$ and $\widehat{G}_{\rm finite},$ respectively.

1. (i) Let $\beta\,:\, G \to \prod_{i \in I} U(n_i)$ be given by $\beta(g)= \bigoplus_{i \in I} \pi_i(g) $ and let K be the closure of $\beta(G)$ . Then $(K, \beta)$ is a Bohr compactification of G.

2. (ii) Let $\alpha\,:\, G \to \prod_{j \in J} U(n_j)$ be given by $\alpha(g)= \bigoplus_{j \in J} \sigma_j(g) $ and let L be the closure of $\alpha(G)$ . Then $(L, \alpha)$ is a profinite completion of G.

We observe that a more common model for the profinite completion of G is the projective limit $\varprojlim G/H$ , where H runs over the family of the normal subgroups of finite index of G, together with the natural homomorphism $G \to \varprojlim G/H$ (see e.g. [ RZ00 , 2·1·6])

2·2. Extension of representations

We will also use the notion of a projective representation. Let G be a locally compact group. A map $\pi\,:\, G \to U(n)$ is a projective representation of G if the following holds:

$\pi(e)=I$ ,

for all $g_1,g_2\in G,$ there exists $c(g_1 , g_2 )\in \textbf{S}^1 $ such that

\begin{equation*}\pi(g_1 g_2 ) = c(g_1 , g_2 )\pi(g_1 )\pi(g_2 ),\end{equation*}

$\pi$ is Borel measurable.

The map $c\,:\,G \times G \to \textbf{S}^1$ is a 2-cocycle with values in the unit circle $\textbf{S}^1.$ The conjugate representation $\overline{\pi}\,:\, G\to U(n)$ is another projective representation defined by $\overline{\pi}(g)= J\pi(g) J,$ where $J\,:\, \textbf{C}^n\to \textbf{C}^n$ is the anti-linear map given by conjugation of the coordinates,

The proof of the following lemma is straightforward.

Lemma 5. Let $\pi\,:\, G\to U(n)$ be a projective representation of G, with associated cocycle $c\,:\,G \times G \to \textbf{S}^1$ . Let $\pi'\,:\, G\to U(m)$ be another projective representation of G with associated cocycle 2-cocycle $c'\,:\,G \times G \to \textbf{S}^1$ .

1. (i) $\overline{\pi}\,:\, G\to U(n)$ is a projective representation of G with $\overline{c}$ as associated cocycle.

2. (ii) The tensor product

   \begin{equation*}\pi\otimes \pi'\,:\, G\to U(nm), \qquad g\mapsto \pi(g)\otimes \pi'(g)\end{equation*}
   is a projective representation of G with cc’ as associated cocycle.

Let N be a closed normal subgroup of G. Recall that the stabiliser ${G}_\pi$ in G of an irreducible unitary representation $\pi$ of N is the set of $g\in G$ such that $\pi^g$ is equivalent to $\pi.$ Observe that $G_\pi$ contains N.

The following proposition is a well known fact from the Clifford–Mackey theory of unitary representations of group extensions (see [ CR62 , chapter 1, section 11] and [ Mac58 ]).

Proposition 6. Let $G= N\rtimes H$ be the semi-direct product of the locally compact groups H and N. Let $\pi\,:\, N\to U(m)$ be an irreducible unitary representation of N and assume that $G= G_\pi.$ There exists a projective representation $\widetilde\pi\,:\, G\to U(m)$ with the following properties:

1. (i) $\widetilde\pi$ extends $\pi$ , that is, $\widetilde\pi(n)= \pi(n)$ for every $n\in N;$

2. (ii) the 2-cocycle $\widetilde{c}\,:\,G \times G\to \textbf{S}^1$ associated to $\widetilde\pi$ has the form $\widetilde{c}=c\circ (p\times p),$ for a map $c\,:\, H\times H\to \textbf{S}^1$ , where $p\,:\, G \to H$ is the canonical homomorphism.

Proof. Let $S\subset U(m)$ be a Borel transversal for the quotient space $PU(m)= U(m)/\textbf{S}^1$ with $I_m\in S.$ Let $h\in H$ . Since $G=G_\pi$ and since $\pi$ is irreducible, there exists a unique matrix $\widetilde\pi(h)\in S$ such that

\begin{equation*}\pi (hn h^{-1})= \widetilde\pi(h) \pi(n) \widetilde\pi(h)^{-1} \qquad\text{for all}\quad n\in N.\end{equation*}

Define $\widetilde\pi\,:\, G\to U(n)$ by

\begin{equation*}\widetilde\pi(nh)= \pi(n)\widetilde\pi(h) \qquad\text{for all}\quad n\in N, h\in H.\end{equation*}

It is clear that $\widetilde{\pi}|_N= \pi$ and that

\begin{equation*}\pi (gng^{-1})= \widetilde\pi(g) \pi(n) \widetilde\pi(g)^{-1} \qquad\text{for all}\quad g\in G, n\in N.\end{equation*}

It can be shown (see [ Mac58 , proof of theorem 8·2]) that $\widetilde{\pi}$ is a measurable map.

Let $g_1,g_2\in G$ . For every $n\in N,$ we have, on the one hand,

\begin{equation*}\pi (g_1g_2n g_2^{-1}g_1)=\widetilde\pi(g_1 g_2)\pi(n) \widetilde\pi(g_1 g_2)^{-1}\end{equation*}

and on the other hand

\[\begin{aligned}\pi (g_1g_2n g_2^{-1}g_1)&= \widetilde\pi(g_1) \pi(g_2 n g_2^{-1}) \widetilde\pi(g_1)^{-1}\\&=\widetilde\pi(g_1) \widetilde\pi(g_2)\pi(n)\widetilde\pi (g_1)^{-1} \widetilde\pi (g_2)^{-1}.\end{aligned}\]

Since $\pi$ is irreducible, it follows that

\begin{equation*}\widetilde\pi(g_1 g_2)=\widetilde{c}(g_1, g_2) \widetilde\pi (g_1)\widetilde\pi(g_2)\end{equation*}

for some scalar $\widetilde{c}(g_1, g_2)\in \textbf{S}^1$ .

Moreover, for $g_1= n_1h_1, g_2= n_2 h_2,$ we have, on the one hand,

\[ \begin{aligned} \widetilde\pi(g_1 g_2)&= \widetilde{c}(g_1, g_2) \widetilde\pi (g_1)\widetilde\pi(g_2)\\ &= \widetilde{c}(n_1 h_1, n_2 h_2) \pi(n_1)\widetilde\pi(h_1) \pi(n_2)\widetilde\pi(h_2) \end{aligned} \]

and, on the other hand,

\[ \begin{aligned} \widetilde\pi(g_1 g_2)&=\widetilde\pi(n_1(h_1n_2 h_1^{-1}) h_1h_2)\\ & =\pi(n_1(h_1n_2 h_1^{-1})) \widetilde \pi(h_1h_2)\\ &=\pi(n_1) \pi(h_1n_2 h_1^{-1}) \widetilde \pi(h_1h_2)\\ &=\pi(n_1) \widetilde\pi(h_1) \pi(n_2) \widetilde\pi(h_1)^{-1} \widetilde \pi(h_1h_2)\\ &=\widetilde{c}(h_1, h_2) \pi(n_1) \widetilde\pi(h_1) \pi(n_2) \widetilde\pi(h_1)^{-1} \widetilde \pi(h_1)\widetilde \pi(h_2)\\ &=\widetilde{c}(h_1, h_2)\pi(n_1) \widetilde\pi(h_1) \pi(n_2)\widetilde \pi(h_2); \end{aligned} \]

this shows that $\widetilde{c}(n_1 h_1, n_2 h_2)=\widetilde{c}(h_1, h_2).$

2·3. Bohr compactification and profinite completion of quotients

Let G be a topological group and N a closed normal subgroup of G. Let $({\rm Bohr}(G), \beta_G)$ and $({\rm Prof}(G), \alpha_G)$ be a Bohr compactification and a profinite completion of G. Let ${\rm Bohr}(p)\,:\, {\rm Bohr}(G)\to {\rm Bohr}(G/N)$ and ${\rm Prof}(p)\,:\, {\rm Bohr}(G)\to {\rm Bohr}(G/N)$ be the morphisms induced by the canonical epimorphism $p\,:\, G\to G/N$ . The following proposition is well known (see [ HK01 , lemma 2·2] or [ Bek23 , proposition 10] for (i) and [ RZ00 , proposition 3·2·5] for (ii)). For the convenience of the reader, we give for (ii) a proof which is different from the one in [ RZ00 ]

Proposition 7

1. (i) $\mathrm{Bohr}(p)$ is surjective and its kernel is $\overline{\beta_G(N)}$ .

2. (ii) $\mathrm{Prof}(p)$ is surjective and its kernel is $\overline{\alpha_G(N)}$ .

Proof. To show (ii), set $K\,:\!=\,\overline{\alpha_G(N)}$ . Let $({\rm Prof}(G/N),\overline{\alpha})$ be a profinite completion of $G/N.$ We have a commutative diagram

It follows that $\alpha_G(N)$ and hence K is contained in ${{\rm Ker}}({\rm Prof}(p)).$ So, we have induced homomorphisms $\beta\,:\, G/N\to {\rm Prof}(G)/K$ and $\beta'\,:\, {\rm Prof}(G)/K\to {\rm Prof}(G/N),$ giving rise to a commutative diagram

It follows that $({\rm Prof}(G)/K, \beta)$ has the same universal property for $G/N$ as $({\rm Prof}(G/N),\overline{\alpha})$ ; it is therefore a profinite completion of $G/N.$

3. Proof of Theorems A and B

3·1. Proof of Theorem A

Set $K\,:\!=\, \overline{\beta_G(N)},$ where $\beta_G$ is the canonical map from the locally compact group $G= N\rtimes H$ to ${\rm Bohr}(G).$

1. (i) First step. We claim that

   \begin{equation*} \left\{\widehat{\sigma}\circ (\beta_G|_N)\,:\, \widehat{\sigma} \in \widehat{K}\right\} \subset \widehat{N}_{ \rm fd}^{H-{\rm per}}. \end{equation*}

Indeed, let $\widehat{\sigma} \in \widehat{K}$ . Then $\sigma\,:\!=\, \widehat{\sigma}\circ (\beta_G|_N)\in \widehat{N}_{ \rm fd}.$ Let $ \widehat{\rho}\in \widehat{{\rm Bohr}(G)}$ be an irreducible subrepresentation of the induced representation ${\rm Ind}_{K}^{{\rm Bohr}(G)} \widehat{\sigma}.$ Then, by Frobenius reciprocity, $\widehat{\sigma}$ is equivalent to a subrepresentation of $\widehat{\rho}|_K.$ Hence, $\sigma$ is equivalent to a subrepresentation of $(\widehat{\rho} \circ \beta_G)|N.$ The decomposition of the finite dimensional representation $(\widehat{\rho} \circ \beta_G)|_N$ into isotypical components shows that $\sigma$ has a finite H-orbit (see [ Bek23 , proposition 12]).

1. (ii) Second step. We claim that

   \begin{equation*} \widehat{N}_{ \rm fd}^{H-{\rm per}} \subset \left\{\widehat{\sigma} \circ (\beta_G|_N)\,:\, \widehat{\sigma} \in \widehat{K}\right\}. \end{equation*}

Indeed, let $\sigma\,:\, N\to U(m)$ be a representation of N with finite H-orbit. By Proposition 6, there exists a projective representation $\widetilde{\sigma}$ of $G_\sigma=NH_\sigma$ which extends $\sigma$ and the associated cocycle $c\,:\, G_\sigma\times G_\sigma\to \textbf{S}^1$ , factorises through $H_\sigma\times H_\sigma$ .

Define a projective representation $\tau\,:\, G_\sigma\to U(m)$ of $G_\sigma$ by

\begin{equation*} \tau(nh)= \overline{\widetilde{\sigma}}(h) \qquad\text{for all}\quad nh\in NH_\sigma. \end{equation*}

Observe that $\tau$ is trivial on N and that its associated cocycle is $\overline{c}.$ Consider the tensor product representation $\widetilde{\sigma}\otimes \tau$ of $G_\sigma.$ Lemma 5 shows that $\widetilde{\sigma}\otimes \tau$ is a projective representation for the cocyle $c\overline{c}=1.$ So, $\widetilde{\sigma}\otimes \tau$ is a measurable homomorphism from $G_\sigma$ to $U(m).$ This implies that $\widetilde{\sigma}\otimes \tau$ is continuous (see [ BHV08 , lemma A·6·2]) and so $\widetilde{\sigma}\otimes \tau$ is an ordinary representation of $G_\sigma$ .

It is clear that $\widetilde{\sigma}\otimes \tau$ is finite dimensional. Observe that the restriction $(\widetilde{\sigma}\otimes \tau)|_N$ of $\widetilde{\sigma}\otimes \tau$ to N is a multiple of $\sigma.$ Let

\begin{equation*}\rho\,:\!=\,{\rm Ind}_{G_\sigma}^G (\widetilde{\sigma}\otimes \tau).\end{equation*}

Then $\rho$ is finite dimensional, since $\widetilde{\sigma}\otimes \tau$ is finite dimensional and $G_\sigma$ has finite index in G. As $G_\sigma$ is open in G, $\widetilde{\sigma}\otimes \tau$ is equivalent to a subrepresentation of the restriction $\rho|_{G_\sigma}$ of $\rho$ to $G_\sigma$ (see e.g. [ BdlH , 1·F]); consequently, $\sigma$ is equivalent to a subrepresentation of $\rho|_{N}$ . Since $\rho$ is a finite dimensional unitary representation of G, there exists a unitary representation $\widehat{\rho}$ of ${\rm Bohr}(G)$ such that $\widehat{\rho}\circ \beta_G= \rho.$ So, $\sigma$ is equivalent to a subrepresentation of $(\widehat{\rho}\circ \beta_G)|_N$ , that is, there exists a subspace V of the space of $\widehat{\rho}$ which is invariant under $\beta_G(N)$ and defining a representation of N which is equivalent to $\sigma.$ Then V is invariant under $K=\overline{\beta_G(N)}$ and defines therefore an irreducible representation $\widehat{\sigma}$ of K for which $\widehat{\sigma}\circ (\beta_G|_N)= \sigma$ holds.

Let $\varphi_N\,:\, {\rm Bohr}(N)\to K=\overline{\beta_G(N)}$ be the homomorphism such that $\varphi_N\circ \beta_N= \beta_G|_N$ .

1. (iii) Third step. We claim that

   \begin{equation*}{{\rm Ker}} \varphi_N =\bigcap_{\sigma \in \widehat{N}_{ \rm fd}^{H-{\rm per}}} {{\rm Ker}} ({\rm Bohr}({\sigma})),\end{equation*}
   where ${\rm Bohr}({\sigma})$ is the representation of ${\rm Bohr}(N)$ such that ${\rm Bohr}({\sigma})\circ \beta_N= \sigma.$

Indeed, by the first and second steps, we have

\begin{equation*}\widehat{N}_{ \rm fd}^{H-{\rm per}}=\left\{\widehat{\sigma} \circ (\beta_G|_N)\,:\, \widehat{\sigma} \in \widehat{K}\right\}=\left\{(\widehat{\sigma} \circ \varphi_N) \circ \beta_N\,:\, \widehat{\sigma} \in \widehat{K}\right\};\end{equation*}

since obviously $\widehat{\sigma} \circ \varphi_N= {\rm Bohr}({\sigma})$ for $\sigma= (\widehat{\sigma} \circ \varphi_N) \circ \beta_N,$ it follows that

\[\begin{aligned}\bigcap_{\sigma \in \widehat{N}_{ \rm fd}^{H-{\rm per}}} {{\rm Ker}} ({\rm Bohr}({\sigma}))&= \bigcap_{\widehat{\sigma} \in \widehat{K}} {{\rm Ker}} (\widehat{\sigma} \circ \varphi_N). \end{aligned} \]

As $\varphi_N({\rm Bohr}(N))=K$ and $\widehat{K}$ separates the points of K, we have $\bigcap_{\widehat{\sigma} \in \widehat{K}} {{\rm Ker}} (\widehat{\sigma} \circ \varphi_N)={{\rm Ker}} \varphi_N$ and the claim is proved.

Set $L\,:\!=\, \overline{\beta_G(H)}.$

1. (iv) Fourth step. We claim that the map $\varphi_H\,:\, {\rm Bohr}(H)\to L,$ defined by the relation $\varphi_H\circ \beta_H= \beta_G|_H,$ is an isomorphism. Indeed, the canonical isomorphism $H\to G/N$ induces an isomorphism ${\rm Bohr}(H) \to {\rm Bohr}(G/N)$ . Using Proposition 7 (i), we obtain a continuous epimorphism

   \begin{equation*}f\,:\,L\to {\rm Bohr}(H)\end{equation*}
   such that $f(\beta_G(h))= \beta_H(h)$ for all $h\in H.$ Then $\varphi_H\circ f$ is the identity on $\beta_G(H)$ and hence on L, by density. This implies that $\varphi_H$ is an isomorphism.

Observe that, by the universal property of ${\rm Bohr}(N),$ every element $h\in H$ defines a continuous automorphism $\theta_b(h)$ of ${\rm Bohr}(N) $ such that

\begin{equation*}\theta_b(h)(n)= \beta_N(hnh^{-1})\qquad\text{for all}\quad n\in N.\end{equation*}

The corresponding homomorphism $\theta_b\,:\,H\to {{\rm Aut}}({\rm Bohr}(N))$ defines an action of H on the compact group ${\rm Bohr}(N).$ By duality, we have an action, still denoted by $\theta_b,$ of H on $\widehat{{\rm Bohr}(N)}$ and we have

\begin{equation*} {\rm Bohr}(\sigma^h)= \theta_b(h)({\rm Bohr}(\sigma)) \qquad\text{for all}\quad \sigma \in \widehat{N}_{ \rm fd}, h\in H. \end{equation*}

This implies that the normal subgroup

\begin{equation*}{{\rm Ker}} \varphi_N=\bigcap_{\sigma \in \widehat{N}_{ \rm fd}^{H-{\rm per}}} {{\rm Ker}} ({\rm Bohr}(\sigma))\end{equation*}

of ${\rm Bohr}(N)$ is H-invariant. We have therefore an induced action $\overline{\theta_b}$ of H on ${\rm Bohr}(N)/{{\rm Ker}}\varphi_N.$ Observe that the isomorphism

\begin{equation*}{\rm Bohr}(N)/{{\rm Ker}}\varphi_N\to K\end{equation*}

induced by $\varphi_N$ is H-equivariant for $\overline{\theta_b}$ and the action of H on K given by conjugation with $\beta_G(h)$ for $h\in H.$

1. (v) Fifth step. We claim that the action $\overline{\theta_b}$ induces an action of ${\rm Bohr}(H)$ by automorphisms on ${\rm Bohr}(N)/{{\rm Ker}} \varphi_N$ and that the map

   \begin{equation*}({\rm Bohr}(N)/{{\rm Ker}}\varphi_N) \rtimes {\rm Bohr}(H)\to {\rm Bohr}(G), (x {{\rm Ker}}\varphi_N, y) \mapsto \varphi_N(x)\varphi_H(y)\end{equation*}
   is an isomorphism.

Indeed, $\overline{\beta_G(N)}$ is a normal subgroup of ${\rm Bohr}(G)$ and so $\overline{\beta_G(H)}$ acts by conjugation on K. By the third and the fourth step, the maps

\begin{equation*}\overline{\varphi_N}\,:\,{\rm Bohr}(N)/{{\rm Ker}} \varphi_N \to K, \qquad x {{\rm Ker}}\varphi_N\mapsto \varphi_N(x)\end{equation*}

and

\begin{equation*} \varphi_H:{\rm Bohr}(H)\to L\end{equation*}

are isomorphisms. We define an action

\begin{equation*}\widehat{\theta}\,:\,{\rm Bohr}(H) \to {{\rm Aut}}({\rm Bohr}(N)/{{\rm Ker}} \varphi_N)\end{equation*}

by

\begin{equation*}\widehat{\theta}(y)(x {{\rm Ker}}\varphi_N)= (\overline{\varphi_N})^{-1} \left(\varphi_H(y) \varphi_N(x) \varphi_H(y)^{-1}\right)\end{equation*}

for $x\in {\rm Bohr}(N)$ and $y\in {\rm Bohr}(H).$ The claim follows.

3·2. Proof of Theorem B

The proof is similar to the proof of Theorem A. The role of $\widehat{N}_{ \rm fd}$ is now played by the space $\widehat{N}_{ \rm finite}$ of finite dimensional irreducible representations of N with finite image. We will go quickly through the steps of the proof of Theorem A; at some places (especially the second step) there will be a few crucial changes and new arguments which we will emphasise.

Set $L\,:\!=\, \overline{\alpha_G(N)},$ where $\alpha_G\,:\,G \to {\rm Prof}(G)$ is the canonical map. Observe that L is profinite.

1. (i) First step. We claim that $ \left\{\widehat{\sigma}\circ (\alpha_G|_N)\,:\,\widehat{\sigma} \in \widehat{L}\right\} \subset \widehat{N}_{ \rm finite}^{H-{\rm per}}. $ Indeed, let $\widehat{\sigma} \in \widehat{L}$ . Then $\sigma\,:\!=\, \widehat{\sigma}\circ (\alpha_G|_N)\in \widehat{N}_{ \rm finite},$ since L is profinite. Let $ \widehat{\rho}$ be an irreducible subrepresentation of ${\rm Ind}_{L}^{{\rm Prof}(G)} \widehat{\sigma}.$ Since ${\rm Prof}(G)$ is compact, $\widehat{\rho}$ is finite dimensional. Since $\sigma$ is equivalent to a subrepresentation of $\widehat{\rho} \circ (\alpha_G)|N)$ , it has therefore a finite H-orbit.

2. (ii) Second step. We claim that $ \widehat{N}_{ \rm finite}^{H-{\rm per}} \subset \left\{\widehat{\sigma} \circ (\alpha_G|_N)\,:\,\widehat{\sigma} \in \widehat{L}\right\}. $ Indeed, let $\sigma\,:\,N\to U(m)$ be an irreducible representation with finite image. By Proposition 6, there exists a projective representation $\widetilde{\sigma}$ of $G_\sigma=NH_\sigma$ which extends $\sigma$ and the associated cocycle $c\,:\,G_\sigma\times G_\sigma\to \textbf{S}^1$ , factorises through $H_\sigma\times H_\sigma$ . We need to show that we can choose $\widetilde{\sigma}$ so that $\widetilde{\sigma}(G_\sigma)$ is finite.

Choose a projective representation $\widetilde{\sigma}\,:\,G_\sigma \to U(m)$ as above and modify $\widetilde{\sigma}$ as follows: define

\begin{equation*}\widetilde{\sigma}_1(nh)= \dfrac{1}{(\det \widetilde{\sigma}(h))^{1/m}} \widetilde{\sigma}(h) \sigma(n) \qquad\text{for all}\quad n\in N, \ h\in H_\sigma. \end{equation*}

Then $\widetilde{\sigma}_1$ is again a projective representation of $G_\sigma=NH_\sigma$ which extends $\sigma$ and the associated cocycle $c\,:\,G_\sigma\times G_\sigma\to \textbf{S}^1$ factorises through $H_\sigma\times H_\sigma$ ; moreover, $\widetilde{\sigma}_1(h)\in SU(m)$ for every $h\in H_\sigma.$

Every $h\in H_\sigma$ induces a bijection $\varphi_h$ of $\sigma(N)$ given by

\begin{equation*}\varphi_h\,:\,\sigma(n) \mapsto \widetilde{\sigma}_1(h)\sigma(n) \widetilde{\sigma}_1(h)^{-1}= \sigma (h nh^{-1}) \qquad\text{for all}\quad n\in N.\end{equation*}

So, we have a map

\begin{equation*}\varphi:\widetilde{\sigma}_1(H_\sigma)\to{\rm Sym}(\sigma(N)), \quad \widetilde{\sigma}_1(h)\mapsto\varphi_h,\end{equation*}

where ${\rm Sym}(\sigma(N))$ is the set of bijections of $\sigma(N).$ For $h_1, h_2\in H_\sigma,$ we have $\varphi_{h_1}= \varphi_{h_2}$ if and only if $\widetilde{\sigma}_1(h_2)= \lambda \widetilde{\sigma}_1(h_1)$ for some scalar $ \lambda \in \textbf{S}^1,$ by irreducibility of $\sigma$ . Since $\det(\widetilde{\sigma}_1(h_1))=1$ and $\det (\widetilde{\sigma}_1(h_2))=1,$ it follows that $\lambda$ is a mth root of unity. This shows that the fibers of the map $\varphi$ are finite. Since $\sigma(N)$ is finite, ${\rm Sym}(\sigma(N))$ and hence $\widetilde{\sigma}_1(H_\sigma)$ is finite. It follows that $\widetilde{\sigma}_1(G_\sigma)= \widetilde{\sigma}_1(H_\sigma)\sigma(N)$ is finite.

Let $\tau\,:\,G_\sigma\to U(m)$ be the projective representation of $G_\sigma$ given by

\begin{equation*} \tau(nh)= \overline{\widetilde{\sigma}_1}(h) \qquad\text{for all}\quad nh\in NH_\sigma. \end{equation*}

Then $\widetilde{\sigma}_1\otimes \tau$ is a ordinary representation of $G_\sigma$ and has finite image. The induced representation $\rho\,:\!=\,{\rm Ind}_{G_\sigma}^G (\widetilde{\sigma}_1\otimes \tau)$ has finite image, since $G_\sigma$ has finite index in G. As $\widetilde{\sigma}_1\otimes \tau$ is equivalent to a subrepresentation of the restriction $\rho|_{G_\sigma}$ of $\rho$ to $G_\sigma,$ the representation $\sigma$ is equivalent to a subrepresentation of $\rho|_{N}$ . Since $\rho(G)$ has finite image, there exists a unitary representation $\widehat{\rho}$ of ${\rm Prof}(G)$ such that $\widehat{\rho}\circ \alpha_G= \rho.$ So, there exists a subspace V of the space of $\widehat{\rho}$ which is invariant under $\alpha_G(N)$ and defining a representation of N which is equivalent to $\sigma.$ Then V defines an irreducible representation $\widehat{\sigma}$ of L for which $\widehat{\sigma}\circ (\alpha_G|_N)= \sigma$ holds.

Let $\psi_N\,:\,{\rm Prof}(N)\to L$ be the homomorphism such that $\psi_N\circ \alpha_N= \alpha_G|_N$ .

1. (iii) Third step. We claim that

   \begin{equation*}{{\rm Ker}} \psi_N =\bigcap_{\sigma \in \widehat{N}_{ \rm finite}^{H-{\rm per}}} {{\rm Ker}} ({\rm Prof}(\sigma)).\end{equation*}
   Indeed, the proof is similar to the proof of the third step of Theorem A

2. (iv) Fourth step. We claim that the map $\psi_H\,:\,{\rm Prof}(H)\to \overline{\alpha_G(H)},$ defined by the relation $\varphi_H\circ \alpha_H= \alpha_G|_H,$ is an isomorphism. Indeed, the proof is similar to the proof of the fourth step of Theorem A.

Every element $h\in H$ defines a continuous automorphism $\theta_p(h)$ of ${\rm Prof}(N).$ Let

\begin{equation*}\theta_p\,:\,H\to {{\rm Aut}}({\rm Prof}(N))\end{equation*}

be the corresponding homomorphism; as in Theorem A, we have an induced action $\overline{\theta_p}$ of H on ${\rm Prof}(N)/{{\rm Ker}}\psi_N.$

• • Fifth step. We claim that the action $\overline{\theta_p}$ of H induces an action of ${\rm Prof}(H)$ by automorphisms on ${\rm Prof}(N)/{{\rm Ker}} \psi_N$ and that the map

  \begin{equation*}\left({\rm Prof}(N)/{{\rm Ker}}\psi_N\right) \rtimes {\rm Prof}(H)\to {\rm Prof}(G), (x {{\rm Ker}}\psi_N, y) \mapsto \psi_N(x)\psi_H(y)\end{equation*}
  is an isomorphism.

Indeed, the proof is similar to the proof of the fifth step of Theorem A.

4. Proof of the Corollaries

4·1. Proof of Corollary C

Assume that N is finitely generated. In view of Theorem B, we have to show that $\widehat{N}_{ \rm finite}^{H-{\rm per}}= \widehat{N}_{ \rm finite}.$

It is well known that, for every integer $n\geq1,$ there are only finitely many subgroups of index n in N. Indeed, since N is finitely generated, there are only finitely many actions of N on the set $\{1, \dots, n\}.$ Every subgroup M of index n defines an action of N on $N/M$ and hence on $\{1, \dots, n\}$ for which the stabiliser of, say, 1 is M. So, there are only finitely many such subgroups M.

Let $\sigma \in \widehat{N}_{ \rm finite}$ and set $n\,:\!=\, |\sigma(N)|.$ Consider $N_\sigma= \cap_{ M} M,$ where M runs over the subgroups of N of index n. Then $N_\sigma$ is a normal subgroup of N of finite index and, for every $h\in H,$ the representation $\sigma^h$ factorises to a representation of $N/N_\sigma.$ Since $N/N_\sigma$ is a finite group, it has only finitely many non equivalent irreducible representations and the claim is proved.

4·2. Proof of Corollary D

We assume that N is abelian. The dual group of ${\rm Bohr}(N)$ is $\widehat{N}$ and the dual of ${\rm Prof}(N)$ is $\widehat{N}_{\rm finite}$ , viewed as discrete groups. With the notation as in Theorems A and B, the subgroups C and D are respectively the annihilators in ${\rm Bohr}(N)$ and in ${\rm Prof}(N)$ of the closed subgroups $\widehat{N}^{H-{\rm per}}$ and $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ . Hence, ${\rm Bohr}(N)/C$ and ${\rm Prof}(N)/D$ are the dual groups of $\widehat{N}^{H-{\rm per}}$ and $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ , viewed as discrete groups. So, the claim follows from Theorems A and B.

4·3 Proof of Corollary E

In view of Theorems A and B, G is MAP, respectively RF, if and only if

\begin{equation*}{{\rm Ker}} (\varphi_N \circ \beta_N)=\{e\} \qquad \text{and} \qquad {{\rm Ker}} (\varphi_H \circ \beta_H)=\{e\},\end{equation*}

respectively

\begin{equation*}{{\rm Ker}} (\psi_N \circ \alpha_N)=\{e\} \qquad \text{and} \qquad {{\rm Ker}} (\psi_H \circ \alpha_H)=\{e\}.\end{equation*}

So, G is MAP, respectively RF, if and only if

\begin{equation*}\beta_N^{-1}(C)=\{e\} \qquad \text{and} \qquad {{\rm Ker}} ( \beta_H)=\{e\},\end{equation*}

respectively

\begin{equation*}\alpha_N^{-1}(D)=\{e\} \qquad \text{and} \qquad {{\rm Ker}} (\alpha_H)=\{e\}.\end{equation*}

This exactly means that G is MAP, respectively RF, if and only if $\widehat{N}_{ \rm fd}^{H-{\rm per}}$ separates the points of N and H is MAP, respectively $\widehat{N}_{ \rm finite}^{H-{\rm per}}$ separates the points of N and H is RF.

4·4. Proof of Corollary F

We assume that $G= \Lambda\wr_X H$ is the wreath product of the groups $\Lambda$ and H given by a transitive action $H\curvearrowright X;$ set $N\,:\!=\,\oplus_{x\in X}\Lambda.$

1. (a) Assume that X is finite. Then, of course, $\widehat{N}_{\rm fd}^{H-{\rm per}}=\widehat{N}_{\rm fd}$ and $\widehat{N}_{ \rm finite}^{H-{\rm per}}=\widehat{N}_{\rm finite};$ so, the subgroups C and D from Theorems A and B are trivial. Since ${\rm Bohr}(N)=\oplus_{x\in X}{\rm Bohr}(\Lambda)$ and $ {\rm Prof}(N)= \oplus_{x\in X}{\rm Prof}(\Lambda),$ we have

   \[\begin{aligned}&{\rm Bohr}(\Lambda\wr_X H)\cong \left( \oplus_{x\in X}{\rm Bohr}(\Lambda)\right) \rtimes {\rm Bohr}(H) \ \text{and}\\&{\rm Prof}(\Lambda\wr_X H)\cong \left( \oplus_{x\in X}{\rm Prof}(\Lambda)\right) \rtimes {\rm Prof}(H).\end{aligned}\]

2. (b) Assume that X is infinite.

1. (i) First step. We claim that, for every $\sigma \in \widehat{N}_{\rm fd}^{H-{\rm per}},$ we have $\dim \sigma =1$ , that is, $\sigma (N)\subset U(1)=\textbf{S}^1.$

Indeed, assume by contradiction that $\dim \sigma >1$ . Let $\mathcal{F}$ be the family of finite subsets of X. For every $F\in \mathcal{F},$ let N(F) be the normal subgroup of N given by

\begin{equation*}N(F)\,:\!=\,\oplus_{x\in F}\Lambda\end{equation*}

The restriction $\sigma|_{N(F)}$ of $\sigma$ to N(F) has a decomposition into isotypical components:

\begin{equation*} \sigma|_{N(F)} = \oplus_{\pi\in \Sigma_F} n_\pi \pi, \end{equation*}

where $\Sigma_F$ is a (finite) subset of $ \widehat{N(F)}_{\rm fd}$ and the $n_\pi$ ’s some positive integers. As is well known (see, e.g., [ Wei40 , section 17]), every representation in $\widehat{N(F)}_{\rm fd}$ is a tensor product $\otimes_{h\in F}\rho_h$ of irreducible representations $\rho_h$ of $\Lambda$ ; so, we can view $\Sigma_F$ as subset of $\prod_{x\in F}\widehat{\Lambda}_{\rm fd}.$ If $F\subset F',$ then the obvious map $\prod_{x\in F'}\widehat{\Lambda}_{\rm fd}\to \prod_{x\in F}\widehat{\Lambda}_{\rm fd}$ restricts to a surjective map $\Sigma_{F^{\prime}} \to \Sigma_F$ .

Since $\dim \sigma$ is finite, it follows that there exists $F_0\in \mathcal{F}$ such that

\begin{equation*}\dim \pi =1 \qquad\text{for all}\quad \pi\in \Sigma_F, F\in \mathcal{F} \quad \text{with} \quad F\cap F_0=\emptyset\end{equation*}

and

\begin{equation*}\dim\pi_0>1 \quad \text{for some}\quad \pi_0\in\Sigma_{F_0}.\end{equation*}

For $h\in H$ and $F\in \mathcal{F},$ observe that for the decomposition of $\sigma^h|_{N( h^{-1}F)}$ into isotypical components, we have

\begin{equation*}\sigma^{h}|_{N(h^{-1}F)}= \oplus_{\pi\in \Sigma_{F}} n_\pi \pi.\end{equation*}

So, $\sigma^{h}$ and $\sigma$ are not equivalent if $h^{-1}F_0\cap F_0=\emptyset.$

Since X is infinite, we can choose inductively a sequence $(h_n)_{n\geq 0}$ of elements in H by $h_0=e$ and

\begin{equation*}h_{n+1}^{-1} F_{0} \cap \bigcup_{0\leq m\leq n} h_{m}^{-1} F_{0} =\emptyset \qquad\text{for all}\quad n \geq 0.\end{equation*}

The $\sigma^{h_n}$ ’s are then pairwise not equivalent. This is a contradiction, since $\sigma \in \widehat{N}_{\rm fd}^{H-{\rm per}}.$

Let $p:\Lambda\wr_X H\to \Lambda^{\rm Ab}\wr_X H$ be the quotient map, which is given by

\begin{equation*}p((\lambda_x)_{x\in X}, h)= ((\lambda_x [\Lambda, \Lambda])_{x\in X}, h).\end{equation*}

1. (ii) Second step. We claim that the induced maps

   \begin{equation*}{\rm Bohr}(p)\,:\, {\rm Bohr}(\Lambda\wr_X H)\to {\rm Bohr}( \Lambda^{\rm Ab}\wr_X H)\end{equation*}
   and
   \begin{equation*} {\rm Prof}(p)\,:\,{\rm Prof}(\Lambda\wr_X H)\to {\rm Prof}( \Lambda^{\rm Ab}\wr_X H) \end{equation*}
   are isomorphisms.

Indeed, by the first step, every $\sigma \in \widehat{N}_{\rm fd}^{H-{\rm per}}$ factorises through $ N^{\rm Ab}.$ Hence, by Theorems A and B, [N, N] is contained in $C=\ker \varphi_N$ and [N, N] is contained in $D=\ker \psi_N$ . This means that $\beta_G(\ker p)= \{e\}$ and $\alpha_G(\ker p)= \{e\}$ . The claim follows then from Proposition 7.

4·5. Proof of Corollary G

We assume that $G= \Lambda\wr_X H$ is the wreath product of the groups $\Lambda$ and H given by an action $H\curvearrowright X.$ We assume that $\Lambda$ has at least two elements and, as before, we set $N=\oplus_{x\in X}\Lambda .$

1. (a) Assume that X is finite. Then G is MAP (respectively RF) if and only if $\Lambda$ and H are MAP (respectively RF).

Indeed, $\widehat{N}_{ \rm fd}^{H-{\rm per}}= \widehat{N}_{ \rm fd}$ separates the points of N if and only if $\Lambda$ is MAP and $\widehat{N}_{ \rm finite}^{H-{\rm per}}=\widehat{N}_{ \rm finite}$ separates the points of N if and only if $\Lambda$ is RF. The claim follows then from Corollary E.

1. (b) Assume that X is infinite.

Assume that G is MAP. Then, for every H-orbit Y in X, the wreath product $\Lambda\wr_{Y} H,$ which embeds as subgroup of G, is MAP. Since some Y is infinite, Corollary F implies that $\Lambda$ is abelian. So, we may and will from now assume that $\Lambda$ (and hence N) is abelian.

1. (i) First step. We claim that, if $\widehat{N}^{H-{\rm per}}$ separates the points of N, then $H\curvearrowright X$ is RF.

Indeed, recall that the dual group $\widehat{\Lambda}$ of $\Lambda,$ equipped with the topology of pointwise convergence, is a compact group. The dual group $\widehat{N}$ of N can be identified, as topological group, with the product group $\prod_{x\in X}\widehat{\Lambda},$ endowed with the product topology, by means of the duality

\begin{equation*}\left \langle \prod_{x\in X}\chi_x, \oplus_{x\in X}\lambda_x\right\rangle= \prod_{x\in X}\chi_x(\lambda_x)\quad \text{for all} \quad \prod_{x\in X}\chi_x\in \widehat{N}, \oplus_{x\in X}\lambda_x\in N.\end{equation*}

(Observe that the product on the right hand side is well-defined since $\lambda_x=e$ for all but finitely many $x\in X.$ ) The dual action of H on $\widehat{N}$ is given by

\begin{equation*} \left(\prod_{x\in X}\chi_x\right)^h = \prod_{x\in X}\chi_{h^{-1}x} \qquad\text{for all}\quad h\in H. \end{equation*}

For $\Phi\,:\!=\,\prod_{x\in X}\chi_x \in \widehat{N}$ , we have that $\Phi\in \widehat{N}^{H-{\rm per}}$ if and only if there exists a finite index subgroup $H_\Phi$ of H such that

\begin{equation*}\chi_{hx}=\chi_x \qquad\text{for all}\quad h\in H_\Phi, x\in X.\end{equation*}

Let $x_0, x_1$ be two distinct points from X. By assumption, $\widehat{N}^{H-{\rm per}}$ separates the points of N; equivalently, $\widehat{N}^{H-{\rm per}}$ is dense in $\widehat{N}$ . Since $\Lambda$ has at least two elements, we can find $\chi^0\in \widehat{\Lambda}$ and $\lambda_0\in \Lambda$ with $\chi^0(\lambda_0)\neq 1.$ Define $\Phi_0= \prod_{x\in X}\chi_x\in \widehat{N}$ by $\chi_{x_0}= \chi^0$ and $\chi_x= 1_{\Lambda}$ for $x\neq x_0.$ Set

\begin{equation*}\varepsilon\,:\!=\, \dfrac{1}{2} \left|\chi^0(\lambda_0)-1\right|>0.\end{equation*}

Since $\widehat{N}^{H-{\rm per}}$ is dense in $\widehat{N}$ , we can find $\Phi'= \prod_{x\in X}\chi_x^{\prime}\in \widehat{N}^{H-{\rm per}}$ such that

\begin{equation*} | \chi_{x_0}^{\prime}(\lambda_0)- \chi_{x_0}(\lambda_0)| \leq \varepsilon/2 \quad \text{and}\quad | \chi_{x_1}^{\prime}(\lambda_0)- \chi_{x_1}(\lambda_0)|\leq \varepsilon/2. \end{equation*}

We claim that $H_{\Phi^{\prime}} x_0\neq H_{\Phi^{\prime}} x_1$ , where $H_{\Phi^{\prime}}$ is the stabiliser of $\Phi^{\prime}.$ Indeed, assume by contradiction that $ x_0\in H_{\Phi^{\prime}} x_1.$ Then $\chi_{x_0}^{\prime}= \chi_{x_1}^{\prime}$ and hence

\[\begin{aligned}2\varepsilon&= |\chi^0(\lambda_0)-1|\\&\leq |\chi^0(\lambda_0)-\chi_{x_0}^{\prime}(\lambda_0)|+ |\chi_{x_0}^{\prime}(\lambda_0)- 1|\\ &=|\chi_{x_0}(\lambda_0)-\chi_{x_0}^{\prime}(\lambda_0)|+ | \chi_{x_1}^{\prime}(\lambda_0)- \chi_{x_1}(\lambda_0)| \\ &\leq \varepsilon \end{aligned}\]

and this is a contradiction. Since $H_{\Phi^{\prime}}$ has finite index, we have proved that $H\curvearrowright X$ is RF.

1. (ii) Second step. We claim that, if $H\curvearrowright X$ is RF, then $\widehat{N}^{H-{\rm per}}$ separates the points of N.

Indeed, let $\oplus_{x\in X}\lambda_x\in N\setminus\{e\}.$ Then $F=\{x\in X \,:\,\lambda_x\neq e\}$ is a finite and non-empty subset of X. Let $(\chi^0_x)_{x\in F}$ be a sequence in $ \widehat{\Lambda}$ such that $\prod_{x\in F} \chi^0_x(\lambda_x) \neq 1$ (this is possible, since abelian groups are MAP). Since $H\curvearrowright X$ is RF, we can find a subgroup of finite index L of H so that $Lx \neq Lx^{\prime}$ for all $x, x^{\prime}\in F$ with $x\neq x^{\prime}.$ Define $\Phi= \prod_{x^{\prime}\in X}\chi_{x^{\prime}}\in \widehat{N}$ by

\begin{equation*}\chi_{x^{\prime}}=\begin{cases}\chi^0_x& \text{if}\quad x^{\prime}\in Lx \quad \text{for some} \quad x\in F,\\1_{\Lambda}& \text{if}\quad x^{\prime}\notin \cup_{x\in F} Lh.\end{cases}\end{equation*}

It is clear that $L\subset H_{\Phi}$ and hence that $\Phi\in \widehat{N}^{H-{\rm per}}$ ; moreover,

\begin{equation*}\Phi\left(\oplus_{x\in X}\lambda_x \right)= \prod_{x\in F} \chi^0_x(\lambda_x) \neq 1.\end{equation*}

So, $\widehat{N}^{H-{\rm per}}$ separates the points of N.

1. (iii) Third step. We claim that, if $H\curvearrowright X$ is RF and $\Lambda$ is RF, then $\widehat{N}_{\rm finite}^{H-{\rm per}}$ separates the points of N.

The proof is the same as the proof of the second step, with only one difference: one has to choose a sequence $(\chi^0_x)_{x\in F}$ in $ \widehat{\Lambda}_{\rm finite}$ such that $\prod_{x\in F} \chi^0_x(\lambda_x) \neq 1$ ; this is possible, since we are assuming that $\Lambda$ is RF.

1. (iv) Fourth step. We claim that G is MAP if and only if H is RF and $H\curvearrowright X$ is RF. Indeed, this follows from Corollary E, combined with the first and second steps.

2. (v) Fifth step. We claim that G is RF if and only if $\Lambda,H$ are RF and $H\curvearrowright X$ is RF. Indeed, this follows from Corollary E, combined with the first and third steps.

5. Examples

5·1. Lamplighter group

For $m\geq 1,$ denote by $C_m$ the finite cyclic group $\textbf{Z}/m\textbf{Z}.$ Recall that

\begin{equation*}{\rm Bohr}(\textbf{Z}) \cong {\rm Bohr}(\textbf{Z})_0 \oplus {\rm Prof}(\textbf{Z}).\end{equation*}

and that

\begin{equation*}{\rm Prof}(\textbf{Z})= \varprojlim_{m} C_m \quad \text{and}\quad {\rm Bohr}(\textbf{Z})_0\cong \prod_{ \omega \in \mathfrak{c}} \textbf{A}/\textbf{Q},\end{equation*}

where $\textbf{A}/\textbf{Q}$ is the ring of adeles of $\textbf{Q}$ and $\mathfrak{c}=2^{\aleph_0}$ (see [ Bek23 , proposition 11]).

For an integer $n_0\geq 2,$ let $G= C_{n_0} \wr \textbf{Z}$ be the lamplighter group. We claim that

\begin{equation*}{\rm Bohr}(G)\cong {\rm Bohr}(\textbf{Z})_0 \times {\rm Prof}(G)\end{equation*}

and

\begin{equation*}{\rm Prof}(G)= \varprojlim_{m} C_{n_0} \wr C_m.\end{equation*}

Indeed, let $N\,:\!=\,\oplus_{k\in \textbf{Z}}C_{n_0}.$ It will be convenient to describe N as the set of maps $f\,:\,\textbf{Z}\to C_{n_0}$ such that ${\rm supp} (\,f)\,:\!=\,\{k\in \textbf{Z} \,:\,f(k)\neq 0\}$ is at most finite. The action of $m\in \textbf{Z}$ on $f\in N$ is given by translation: $f^m(k)= f(k+m)$ for all $k\in \textbf{Z}.$

We identify $\widehat{C_{n_0} }$ with the group $\mu_{n_0}$ of $n_0$ -th roots of unity in $\textbf{C}$ by means of the duality

\begin{equation*}\langle z, k\textbf{Z} \rangle= z^k \qquad\text{for all}\quad z\in \mu_{n_0}, k\in \textbf{Z}.\end{equation*}

Then $\widehat{N}$ can be identified with the set of maps $\Phi\,:\,\textbf{Z}\to \mu_{n_0},$ with duality given by

\begin{equation*}\langle \Phi, f \rangle= \prod_{k\in \textbf{Z}}\langle \Phi(k), f(k) \rangle \qquad\text{for all}\quad \Phi\in \widehat{N}, f\in N.\end{equation*}

Observe that $\Phi(N) \subset \mu_{n_0}$ and so $\widehat{N}=\widehat{N}_{\rm finite}.$

We have $\widehat{N}^{H-{\rm per}}=\bigcup_{m\geq 1}\widehat{N}(m),$ where $\widehat{N}(m)$ is the subgroup

\begin{equation*}\widehat{N}(m)=\left \{\Phi\,:\,\textbf{Z}\to \mu_{n_0} \,:\,\Phi (k+m)= \Phi(k) \quad \text{for all}\quad k\in \textbf{Z}\right\}.\end{equation*}

Observe that we have natural injections $i_{m_2}^{m_1}\,:\,\widehat{N}(m_2) \to \widehat{N}(m_1)$ if $m_1$ is a multiple of $m_2.$ The dual group A(m) of $\widehat{N}(m)$ can be identified with the set of maps $\overline{f}\,:\,C_m\to C_{n_0}$ by means of the duality

\begin{equation*} \langle \overline{f}, \Phi \rangle= \prod_{k+m\textbf{Z}\in C_m} \Phi(k)^{\overline{f}(k+m\textbf{Z})} \qquad\text{for all}\quad \Phi\in \widehat{N}(m), \overline{f}\in A(m).\end{equation*}

If $m_1$ is a multiple of $m_2,$ we have a projection $p_{m_1}^{m_2}\,:\,A(m_1) \to A(m_2)$ given by

\begin{equation*}\langle p_{m_1}^{m_2} (\overline{f}), \Phi\rangle= \langle \overline{f}, \Phi\circ i_{m_2}^{m_1} \rangle.\end{equation*}

The dual group A of $\widehat{N}^{H-{\rm per}}=\bigcup_{m\geq 1}\widehat{N}(m)$ can then be identified with the projective limit $\varprojlim_{m} A(m)$ .

The action of $ \textbf{Z}$ by automorphisms of A is given, for $r\in \textbf{Z}$ and $\overline{f}=(\overline{f}_m)_{m\geq 1}\in A$ by $(\overline{f})^r= (\overline{g}_m)_{m\geq 1},$ where

\begin{equation*} \overline{g}_m(k+m\textbf{Z})= \overline{f}_m(k+r+m\textbf{Z}) \qquad\text{for all}\quad k\in \textbf{Z}.\end{equation*}

This action extends to an action of ${{\rm Proj}}(\textbf{Z}) = {\varprojlim}_m C_m$ by automorphisms on A in an obvious way. By Corollary D, the group ${\rm Prof}(G)$ is isomorphic to the corresponding semi-direct product $A \rtimes {\rm Prof} (\textbf{Z})$ and hence

\begin{equation*}{\rm Prof}(G) \cong \varprojlim_m C_{n_0} \wr C_m.\end{equation*}

By Corollary D again, the action of $\textbf{Z}$ on A extends to an action by automorphisms of ${\rm Bohr}(\textbf{Z}).$ Since ${\rm Bohr}(\textbf{Z})_0$ is connected and A is totally disconnected, ${\rm Bohr}(\textbf{Z})_0$ acts as the identity on A. Since ${\rm Bohr}(\textbf{Z})\cong {\rm Bohr}(\textbf{Z})_0 \times {\rm Prof}(\textbf{Z}),$ it follows that

\begin{equation*}{\rm Bohr}(G) \cong (A\rtimes {{\rm Proj}}(\textbf{Z})) \times {{\rm Bohr}}(\textbf{Z})_0 \cong {{\rm Prof}}(G) \times {\rm Bohr}(\textbf{Z})_0.\end{equation*}

For another description of ${\rm Prof}(G),$ see [ GK14 , lemma 3·24].

5·2. Heisenberg group

Let R be a commutative unital ring. The Heisenberg group is the group

\begin{equation*} H(R)\,:\!=\,\left\{ \left( \begin{array}{ccc} 1&a&c\\ 0&1&b\\0&0&1 \end{array}\right)\,:\,a,b, c\in R \right\}.\end{equation*}

We can and will identify H(R) with $R^3$ , equipped with the group law

\begin{equation*}(a,b,c) (a', b', c') \, = \, (a + a', b + b', c + c' + ab').\end{equation*}

We will equip R with the discrete topology; in the sequel, ${\rm Bohr}(R), {\rm Prof}(R),$ and $\widehat{R}$ will be the Bohr compactification, the profinite completion, and the dual group of $(R, +)$ , the additive group of R.

Let $ {\mathcal I}_{\rm finite}$ be the family of ideals of the ring R with finite index (as subgroups of $(R,+)$ ). Every ideal I from ${\mathcal I}_{\rm finite}$ defines two compact groups $H({\rm Bohr}(R), I)$ and $H({\rm Prof}(R), I)$ of Heisenberg type as follows:

\begin{equation*}H({\rm Bohr}(R),I)\,:\!=\, {\rm Bohr}(R)\times {\rm Bohr}(R)\times (R/I)\end{equation*}

is equipped with the group law

\begin{equation*}(x,y,z) (x', y', z') \, = \, (x + x', y + y', z + z' + p_I(x)p_I(y'),\end{equation*}

where $p_I\,:\,{\rm Bohr}(R)\to R/I$ is the group homomorphism induced by the canonical map $R\to R/I;$ the group $H({\rm Prof}(R),I)$ is defined in a similar way.

Observe that, for two ideals I and J in ${\mathcal I}_{\rm finite}$ with $J\subset I,$ we have natural epimorphisms

\begin{equation*}H({\rm Bohr}(R),J)\to H({\rm Bohr}(R),I) \qquad\text{and} \qquad H({\rm Prof}(R),J)\to H({\rm Prof}(R),I).\end{equation*}

We claim that the canonical maps $H(R)\to H({\rm Bohr}(R),I)$ and $H(R)\to H({\rm Prof}(R),I)$ induce isomorphisms

\begin{equation*}{\rm Bohr}(H(R))\cong \varprojlim_{I} H({\rm Bohr}(R), I)\end{equation*}

and

\begin{equation*}{\rm Prof}(H(R))\cong \varprojlim_{I} H({\rm Prof}(R), I),\end{equation*}

where I runs over $ {\mathcal I}_{\rm finite}.$

Indeed, H(R) is a semi-direct product $N\rtimes H$ for

\begin{equation*}N = \{ (0,b,c) \,:\,b, c \in R \} \cong R^2\end{equation*}

and

\begin{equation*}H = \{ (a,0,0) \,:\,a \in R \} \cong R.\end{equation*}

Let $\chi\in \widehat{N}$ . Then $\chi = \chi_{\beta, \psi}$ for a unique pair $(\beta, \psi) \in (\widehat R)^2$ , where $\chi_{\beta, \psi}$ is defined by

\begin{equation*}\chi_{\beta, \psi}(0,b,c) \, = \, \beta(b) \psi(c)\hskip.5cm \text{for} \hskip.2cmb, c \in R.\end{equation*}

For $h = (a,0,0) \in H$ , we have

\begin{equation*}\chi_{\beta, \psi}^{h}(0,b,c) \, = \, \beta(b) \psi(a^{-1}b) \psi(c)\, = \, \chi_{\beta \psi^a, \psi} (0, b, c)\hskip.5cm \text{for} \hskip.2cmb, c \in R ,\end{equation*}

where $\psi^a \in \widehat R$ is defined by $\psi^a(b) = \psi(a^{-1}b)$ for $b \in R$ . It follows that the H-orbit of $\chi_{\beta, \psi}$ is

\begin{equation*}\{\chi_{\beta \psi^a, \psi} \,:\,a \in R \},\end{equation*}

and that the stabiliser of $\chi_{\beta, \psi}$ , which only depends on $\psi$ , is

\begin{equation*}H_\psi \, = \, \{ (a, 0, 0) \mid a \in I_\psi \},\end{equation*}

where $I_\psi$ is the ideal of R defined by

\begin{equation*}I_\psi \, = \, \{ a \in R \mid aR \subset \ker \psi \}.\end{equation*}

Let ${\widehat R}_{\rm per}$ be the subgroup of all $\psi \in\widehat R$ which factorises through a quotient $R/I$ for an ideal $I\in {\mathcal I}_{\rm finite}.$ It follows that

\begin{equation*}{\widehat N}^{H-{\rm per}}=\{\chi_{\beta, \psi} \,:\,\beta \in \widehat R, \psi \in {\widehat R}_{\rm per} \} \cong {\widehat R} \times {\widehat R}_{\rm per}.\end{equation*}

The dual group of ${\widehat R}_{\rm per}$ can be identified with $ \varprojlim_{I} R/I$ , where I runs over ${\mathcal I}_{\rm finite}.$ So, the dual group A of ${\widehat N}^{H-{\rm per}}$ can be identified with $ \varprojlim_{I} {\rm Bohr}(R) \times (R/I).$

The action of ${\rm Bohr}(H)\cong {\rm Bohr}(R)$ on every ${\rm Bohr}(R) \times (R/I)$ is given by

\begin{equation*} x \cdot (y,z)=(y, z + p_I(x)p_I(y')) \qquad\text{for all}\quad x,y \in {\rm Bohr}(R), z\in R/I, \end{equation*}

for the natural map $p_I\,:\,{\rm Bohr}(R)\to R/I$ . This shows that

\begin{equation*}{\rm Bohr}(H(R))\cong \varprojlim_{I} H({\rm Bohr}(R), I).\end{equation*}

Similarly, the dual group B of ${\widehat N_{\rm finite}}^{H-{\rm per}}$ can be identified with $ \varprojlim_{I} {\rm Prof}(R) \times (R/I)$ and we have

\begin{equation*}{\rm Prof}(H(R))\cong \varprojlim_{I} H({\rm Prof}(R), I).\end{equation*}