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 [ Reference BekkaBek23 , propositions 5 and 6].

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. [ Reference Ribes and ZalesskiiRZ00 , 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.

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 [ Reference Curtis and ReinerCR62 , chapter 1, section 11] and [ Reference MackeyMac58 ]).

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 [ Reference MackeyMac58 , 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 [ Reference Hart and KunenHK01 , lemma 2·2] or [ Reference BekkaBek23 , proposition 10] for (i) and [ Reference Ribes and ZalesskiiRZ00 , 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 [ Reference Ribes and ZalesskiiRZ00 ]

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 $G/N.$

3·1. 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 [ Reference BekkaBek23 , 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 [ Reference Bekka, de la Harpe and ValetteBHV08 , 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. [ Reference Bekka and de la HarpeBdlH , 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.

4·1. 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., [ Reference WeilWei40 , 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.

5·1. 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 [ Reference BekkaBek23 , 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 [ Reference Grigorchuk and KravchenkoGK14 , 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*}