which verifies minimality of $\mathcal{C}\mathcal{D}$.

###### Proof of Corollary 3.20.

Recall that a connected group from $\mathsf{CAGp}$ has only connected quotient
groups and a totally disconnected group from $\mathsf{CAGp}$ has only totally
disconnected quotient groups. Thus, by the assumptions of the corollary, the
groups $F(\mathcal{C})$, $F(\mathcal{D})$ have no common quotient group other
than the trivial one and so they are group-disjoint by Lemma 2.4. By virtue of
(3.10) from Proposition 3.19 it follows that, indeed,
$F(\mathcal{C}\mathcal{D})=F(\mathcal{C})F(\mathcal{D})$. ∎

######  Theorem 3.22.

Let $\mathcal{F}\colon\Gamma\curvearrowright X$ be a minimal flow. For
$n\in\mathbb{N}$ let $G_{n}\in\mathsf{CAGp}$ and
$\mathcal{C}_{n}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G_{n})$. Set
$\mathcal{C}=(\mathcal{C}_{n})_{n\in\mathbb{N}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}\left(\prod_{n\in\mathbb{N}}G_{n}\right)$. If the groups
$F(\mathcal{C}_{n})$ are pair-wise group-disjoint then

$F(\mathcal{C})=\prod_{n\in\mathbb{N}}F(\mathcal{C}_{n}).$

###### Proof.

If we interpret the product $\prod_{n\in\mathbb{N}}G_{n}$ in the usual way as
an inverse limit of the groups $\prod_{n=1}^{m}G_{n}$ ($m\in\mathbb{N}$) then
the statement of the theorem follows from Proposition 3.19 and from Theorem
3.7(5).

To be more precise, for $i\leq k$ let
$p_{ik}\colon\prod_{n=1}^{k}G_{n}\to\prod_{n=1}^{i}G_{n}$ be given by
$p_{ik}(g_{n})_{n=1}^{k}=(g_{n})_{n=1}^{i}$ and for $i\in\mathbb{N}$ let
$p_{i}\colon\prod_{n\in\mathbb{N}}G_{n}\to\prod_{n=1}^{i}G_{n}$ be given by
$p_{i}(g_{n})_{n\in\mathbb{N}}=(g_{n})_{n=1}^{i}$. Then
$\prod_{n=1}^{i}G_{n}\stackrel{{\scriptstyle
p_{ik}}}{{\longleftarrow}}\prod_{n=1}^{k}G_{n}$ is an inverse system in
$\mathsf{CAGp}$, $\prod_{n\in\mathbb{N}}G_{n}$ is its inverse limit and
$p_{i}$ ($i\in\mathbb{N}$) are the corresponding limit projections. By virtue
of Lemma 2.21, $p_{i}\mathcal{C}\stackrel{{\scriptstyle
p_{ik}}}{{\longleftarrow}}p_{k}\mathcal{C}$ is an inverse system in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$, $\mathcal{C}$ is its inverse limit
and $p_{i}\colon\mathcal{C}\to p_{i}\mathcal{C}$ ($i\in\mathbb{N}$) are the
corresponding limit projections. Moreover, by Proposition 3.19,

$F(p_{i}\mathcal{C})=F(\mathcal{C}_{1},\dots,\mathcal{C}_{i})=F(\mathcal{C}_{1})\times\dots\times
F(\mathcal{C}_{i})$

for every $i\in\mathbb{N}$. Thus, the identity (3.6) from Lemma 3.15 yields

$\begin{split}F(\mathcal{C})&=\lim_{\longleftarrow}\left(F(\mathcal{C}_{i})\stackrel{{\scriptstyle
p_{ik}}}{{\longleftarrow}}F(\mathcal{C}_{k})\right)=\bigcap_{n\in\mathbb{N}}p_{n}^{-1}F(p_{n}\mathcal{C})\\\
&=\bigcap_{n\in\mathbb{N}}p_{n}^{-1}\left(F(\mathcal{C}_{1})\times\dots\times
F(\mathcal{C}_{n})\right)=\prod_{n\in\mathbb{N}}F(\mathcal{C}_{n}),\end{split}$

as was to be shown. ∎

######  Theorem 3.23.

Let $\Gamma\in\mathsf{LCGp}$, $X$ be a locally compact space and
$G\in\mathsf{CAGp}$. Assume that all $\Gamma$, $X$ and $G$ are second
countable. Let $\mathcal{F}\colon\Gamma\curvearrowright X$ be a minimal flow.
For $n\in\mathbb{N}$ let $\mathcal{C}_{n}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ and assume that the groups $F(\mathcal{C}_{n})$ are
pair-wise group-disjoint. If $\mathcal{C}_{n}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}e$ then there exists an increasing sequence
$(k_{n})_{n\in\mathbb{N}}$ of positive integers such that the series
$\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}$ converges u.c.s. in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ and

$F\left(\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}\right)\supseteq\overline{\sum_{n=1}^{\infty}F(\mathcal{C}_{k_{n}})},$
(3.11)

where the right-hand side of the inclusion above is defined as the closed
subgroup of $G$ generated by the groups $F(\mathcal{C}_{k_{n}})$:

$\overline{\sum_{n=1}^{\infty}F(\mathcal{C}_{k_{n}})}=\overline{\bigcup_{n\in\mathbb{N}}\left(\sum_{l=1}^{n}F(\mathcal{C}_{k_{l}})\right)}.$
(3.12)

######  Remark 3.24.

Observe that the right-hand side of (3.12) can be expressed as a limit in the
hypersemigroup $2^{G}$:

$\overline{\bigcup_{n\in\mathbb{N}}\left(\sum_{l=1}^{n}F(\mathcal{C}_{k_{l}})\right)}=\lim_{n\to\infty}\sum_{l=1}^{n}F(\mathcal{C}_{k_{l}}).$

Before turning to the proof of Theorem 3.23, we prove three auxiliary lemmas.

######  Lemma 3.25.

Under the assumptions of Theorem 3.23, there exists an increasing sequence of
positive integers $(k_{n})_{n\in\mathbb{N}}$ such that the series
$\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}$ converges u.c.s. in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of
$(k_{n})_{n\in\mathbb{N}}$.

###### Proof.

Let us begin by fixing the following objects:

  * •

a local base $(W_{n})_{n\in\mathbb{N}}$ at $e$ in $G$ with
$W_{n+1}+W_{n+1}\subseteq W_{n}$ for every $n\in\mathbb{N}$,

  * •

a sequence $(K_{n})_{n\in\mathbb{N}}$ of compact subsets of $\Gamma$, whose
interiors cover $\Gamma$ and which satisfy $K_{n+1}\supseteq K_{n}$ for every
$n\in\mathbb{N}$,

  * •

a sequence $(C_{n})_{n\in\mathbb{N}}$ of compacts subsets of $X$, whose
interiors cover $X$ and which satisfy $C_{n+1}\supseteq C_{n}$ for every
$n\in\mathbb{N}$.

Since $\mathcal{C}_{n}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}e$ in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, there is an increasing sequence
$(k_{n})_{n\in\mathbb{N}}$ of positive integers such that for every
$n\in\mathbb{N}$, $\mathcal{C}_{k_{n}}(\gamma,x)\in W_{n}$ for all $\gamma\in
K_{n}$ and $x\in C_{n}$. We show that the sequence $(k_{n})_{n\in\mathbb{N}}$
fulfills the condition from the conclusion of the lemma.

Let $(l_{n})_{n\in\mathbb{N}}$ be a subsequence of $(k_{n})_{n\in\mathbb{N}}$.
Then $l_{n}=k_{m(n)}$ for every $n\in\mathbb{N}$, where
$(m(n))_{n\in\mathbb{N}}$ is an increasing sequence of positive integers. (In
particular, $m(n)\geq n$ for every $n\in\mathbb{N}$.) We show that the series
$\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}$ converges u.c.s. in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$; we accomplish this by showing that
$(\sum_{n=1}^{m}\mathcal{C}_{l_{n}})_{m\in\mathbb{N}}$ is a Cauchy sequence in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. To this end, fix a compact set
$F\subseteq\Gamma\times X$ and a neighbourhood $W$ of $e$ in $G$. Choose
$n_{0}\in\mathbb{N}$ large enough so that $F\subseteq K_{n_{0}}\times
C_{n_{0}}$ and $W_{n_{0}}\subseteq W$. Then for all $n\geq n_{0}$,
$p\in\mathbb{N}$ and $(\gamma,x)\in F$,

$\begin{split}\sum_{j=1}^{n+p}\mathcal{C}_{l_{j}}(\gamma,x)-\sum_{j=1}^{n}\mathcal{C}_{l_{j}}(\gamma,x)&=\sum_{j=n+1}^{n+p}\mathcal{C}_{l_{j}}(\gamma,x)=\sum_{j=n+1}^{n+p}\mathcal{C}_{k_{m(j)}}(\gamma,x)\\\
&\in\sum_{j=n+1}^{n+p}W_{m(j)}\subseteq\sum_{j=n+1}^{n+p}W_{j}\subseteq
W_{n}\subseteq W_{n_{0}}\subseteq W.\end{split}$

Thus, $\left(\sum_{n=1}^{m}\mathcal{C}_{l_{n}}\right)_{m\in\mathbb{N}}$ is
indeed a Cauchy sequence in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Since the
group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is complete, it follows that the
series $\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. This proves the lemma. ∎

######  Lemma 3.26.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma$ and $X$ locally compact second countable. Assume that
$(\mathcal{C}_{n})_{n\in\mathbb{N}}$ is a sequence in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ with $\mathcal{C}_{n}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}1$. If the groups $F(\mathcal{C}_{n})$ are pair-wise
group-disjoint and $F(\mathcal{C}_{n})\neq 1$ for every $n\in\mathbb{N}$ then
there exists an increasing sequence $(k_{n})_{n\in\mathbb{N}}$ of positive
integers such that for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of
$(k_{n})_{n\in\mathbb{N}}$, the series
$\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}$ converges u.c.s. in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ and

$F\left(\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}\right)=\mathbb{T}^{1}=\overline{\sum_{n=1}^{\infty}F(\mathcal{C}_{l_{n}})}.$
(3.13)

###### Proof.

First we indicate how to secure the second equality from (3.13). By our
assumptions on the groups $F(\mathcal{C}_{n})$ ($n\in\mathbb{N}$), we have
$F(\mathcal{C}_{i})=\mathbb{T}^{1}$ for at most one $i\in\mathbb{N}$. By
removing the underlying extension $\mathcal{C}_{i}$ from our considerations,
we get $1\neq F(\mathcal{C}_{n})\subsetneq\mathbb{T}^{1}$ for every
$n\in\mathbb{N}$. Thus, for every $n\in\mathbb{N}$ there is an integer
$p(n)\geq 2$ such that $F(\mathcal{C}_{n})=\mathbb{Z}_{p(n)}$. Since the
groups $F(\mathcal{C}_{n})$ are pair-wise group-disjoint, it follows that the
integers $p(n)$ are mutually co-prime. Consequently, if
$(l_{n})_{n\in\mathbb{N}}$ is an arbitrary increasing sequence of positive
integers then

$\overline{\sum_{n=1}^{\infty}F(\mathcal{C}_{l_{n}})}=\overline{\sum_{n=1}^{\infty}\mathbb{Z}_{p(l_{n})}}=\overline{\bigcup_{n\in\mathbb{N}}\sum_{j=1}^{n}\mathbb{Z}_{p(l_{j})}}=\overline{\bigcup_{n\in\mathbb{N}}\mathbb{Z}_{p(l_{1})\dots
p(l_{n})}}=\mathbb{T}^{1}.$

This verifies the second equality from (3.13).

Now we turn to the first equality from (3.13). By Lemma 3.25, we may assume
that the series $\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ for every increasing sequence of positive
integers $(k_{n})_{n\in\mathbb{N}}$. We fix the following objects:

  * •

a base point $z\in X$,

  * •

a basis $(U_{n})_{n\in\mathbb{N}}$ of neighbourhoods of $z$ in $X$ with
$U_{n+1}\subseteq U_{n}$ for every $n\in\mathbb{N}$,

  * •

a basis $(V_{n})_{n\in\mathbb{N}}$ for the topology of $\mathbb{T}^{1}$,

  * •

a sequence $(\widetilde{V}_{n})_{n\in\mathbb{N}}$ of nonempty open subsets of
$\mathbb{T}^{1}$ and a basis $(W_{n})_{n\in\mathbb{N}}$ of neighbourhoods of
$1$ in $\mathbb{T}^{1}$ such that $\widetilde{V}_{n}+W_{n}\subseteq V_{n}$ and
$W_{n+1}+W_{n+1}\subseteq W_{n}$ for every $n\in\mathbb{N}$.

Now we turn to the next step of the proof. Proceeding by induction on $n$, we
shall construct an increasing sequence $(k_{n})_{n\in\mathbb{N}}$ of positive
integers and a non-decreasing sequence $(F_{n})_{n\in\mathbb{N}}$ of finite
subsets of $\Gamma$. Properties of these sequences will be summarized after
the induction step of the construction and their importance will become clear
in the final step of the proof.

Since the integers $p(n)\geq 2$ ($n\in\mathbb{N}$) are mutually co-prime, we
have $p(n)\to\infty$ as $n\to\infty$ and so there is $k_{1}\in\mathbb{N}$ such
that
$F(\mathcal{C}_{k_{1}})\cap\widetilde{V}_{1}=\mathbb{Z}_{p(k_{1})}\cap\widetilde{V}_{1}\neq\emptyset$.
By definition of the functor $F$ from Lemma 3.11, there is
$\gamma_{0}\in\Gamma$ with $T_{\gamma_{0}}z\in U_{1}$ and
$\mathcal{C}_{k_{1}}(\gamma_{0},z)\in\widetilde{V}_{1}$. Set
$F_{1}=\\{\gamma_{0}\\}$. Further, let $n\in\mathbb{N}$ and assume that we
have at our disposal positive integers $k_{1}<\dots<k_{n}$ and finite sets
$F_{1}\subseteq\dots\subseteq F_{n}\subseteq\Gamma$ such that for every
$m=1,\dots,n$, the following conditions hold:

  1. 1.

given $1\leq i\leq m$ and a (possibly empty) set
$B\subseteq\\{1,\dots,m-1\\}$, there is $\gamma\in F_{m}$ with

$T_{\gamma}z\in U_{m}\hskip 8.53581pt\text{and}\hskip
8.53581pt\left(\sum_{l\in
B}\mathcal{C}_{k_{l}}+\mathcal{C}_{k_{m}}\right)(\gamma,z)\in\widetilde{V}_{i},$

  2. 2.

if $m\geq 2$ then $\mathcal{C}_{k_{m}}(\gamma,z)\in W_{m}$ for every
$\gamma\in F_{m-1}$.

Now we define $k_{n+1}$ and $F_{n+1}$. Since $p(n)\to\infty$ and
$\mathcal{C}_{n}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}1$ as
$n\to\infty$, there is $k_{n+1}>k_{n}$ such that

  * •

$F(\mathcal{C}_{k_{n+1}})\cap\widetilde{V}_{i}=\mathbb{Z}_{p(k_{n+1})}\cap\widetilde{V}_{i}\neq\emptyset$
for every $i=1,\dots,n+1$ and

  * •

$\mathcal{C}_{k_{n+1}}(\gamma,z)\in W_{n+1}$ for every $\gamma\in F_{n}$.

Let $B\subseteq\\{1,\dots,n\\}$ and $1\leq i\leq n+1$. By Proposition 3.19,

$F\left(\sum_{l\in
B}\mathcal{C}_{k_{l}}+\mathcal{C}_{k_{n+1}}\right)=\sum_{l\in
B}F(\mathcal{C}_{k_{l}})+F(\mathcal{C}_{k_{n+1}})\supseteq
F(\mathcal{C}_{k_{n+1}})\cap\widetilde{V}_{i}\neq\emptyset.$

Consequently, there is $\gamma_{B}^{i}\in\Gamma$ such that

$T_{\gamma_{B}^{i}}z\in U_{n+1}\hskip 8.53581pt\text{and}\hskip
8.53581pt\left(\sum_{l\in
B}\mathcal{C}_{k_{l}}+\mathcal{C}_{k_{n+1}}\right)(\gamma_{B}^{i},z)\in\widetilde{V}_{i}.$

Let $F_{n+1}$ be the union of $F_{n}$ with all the elements $\gamma_{B}^{i}$
($B\subseteq\\{1,\dots,n\\}$, $1\leq i\leq n+1$). Then the following
conditions hold:

  * •

given $1\leq i\leq n+1$ and $B\subseteq\\{1,\dots,n\\}$, there is $\gamma\in
F_{n+1}$ with

$T_{\gamma}z\in U_{n+1}\hskip 8.53581pt\text{and}\hskip
8.53581pt\left(\sum_{l\in
B}\mathcal{C}_{k_{l}}+\mathcal{C}_{k_{n+1}}\right)(\gamma,z)\in\widetilde{V}_{i},$

  * •

for every $\gamma\in F_{n}$, $\mathcal{C}_{k_{n+1}}(\gamma,z)\in W_{n+1}$.

This finishes the induction step. To summarize, we obtained an increasing
sequence $(k_{n})_{n\in\mathbb{N}}$ of positive integers and a non-decreasing
sequence $(F_{n})_{n\in\mathbb{N}}$ of finite subsets of $\Gamma$ such that
conditions (1) and (2) hold for every $m\in\mathbb{N}$.

We show that the sequence $(k_{n})_{n\in\mathbb{N}}$ satisfies the required
condition. To this end, let $(l_{n})_{n\in\mathbb{N}}$ be a subsequence of
$(k_{n})_{n\in\mathbb{N}}$ and write $l_{n}=k_{t(n)}$ for $n\in\mathbb{N}$.
The convergence of the series $\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}}$ u.c.s.
in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is guaranteed by one of our earlier
arguments in this proof; denote the sum of this series by $\mathcal{C}$. We
show that $F(\mathcal{C})=\mathbb{T}^{1}$. To this end, fix a neighbourhood
$U$ of $z$ in $X$ and an arbitrary nonempty open subset $V$ of $G$; we need to
find $\gamma\in\Gamma$ so that $T_{\gamma}z\in U$ and
$\mathcal{C}(\gamma,z)\in V$. Choose $n\in\mathbb{N}$ with $U_{n}\subseteq U$
and $\overline{V_{n}}\subseteq V$. Clearly, $t(n)\geq n$. Therefore, by
definition of the set $F_{t(n)}$, there is $\gamma\in F_{t(n)}$ such that

$T_{\gamma}z\in U_{t(n)}\hskip 8.53581pt\text{and}\hskip
8.53581pt\left(\sum_{i=1}^{n}\mathcal{C}_{k_{t(i)}}\right)(\gamma,z)\in\widetilde{V}_{n}.$

Moreover, since $\gamma\in F_{t(n)}$, we have
$\mathcal{C}_{k_{t(i)}}(\gamma,z)\in W_{t(i)}$ for every $i>n$. Thus, for
every $p\in\mathbb{N}$,

$\begin{split}\left(\sum_{i=1}^{n+p}\mathcal{C}_{l_{i}}\right)(\gamma,z)&=\left(\sum_{i=1}^{n+p}\mathcal{C}_{k_{t(i)}}\right)(\gamma,z)=\left(\sum_{i=1}^{n}\mathcal{C}_{k_{t(i)}}\right)(\gamma,z)+\sum_{i=n+1}^{n+p}\mathcal{C}_{k_{t(i)}}(\gamma,z)\\\
&\in\widetilde{V}_{n}+W_{t(n+1)}+W_{t(n+2)}\dots+W_{t(n+p)}\\\
&\subseteq\widetilde{V}_{n}+W_{t(n)+1}+W_{t(n)+2}\dots+W_{t(n)+p}\\\
&\subseteq\widetilde{V}_{n}+W_{t(n)}\subseteq\widetilde{V}_{n}+W_{n}\subseteq
V_{n}.\end{split}$

Consequently,
$\mathcal{C}(\gamma,z)=(\sum_{n=1}^{\infty}\mathcal{C}_{l_{n}})(\gamma,z)\in\overline{V_{n}}\subseteq
V$. Since, in addition, $T_{\gamma}z\in U_{t(n)}\subseteq U_{n}\subseteq U$,
the proof is finished. ∎

######  Lemma 3.27.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma$ and $X$ locally compact second countable. Assume that
$\mathcal{A}\subseteq\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is a finite set and
for $n\in\mathbb{N}$ let $\xi_{n}\colon X\to\mathbb{T}^{1}$ be a continuous
map. If $\mathop{\rm co}(\xi_{n})\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}1$ then there exists an increasing sequence
$(k_{n})_{n\in\mathbb{N}}$ of positive integers such that for every
subsequence $(l_{n})_{n\in\mathbb{N}}$ of $(k_{n})_{n\in\mathbb{N}}$, the
series $\sum_{n=1}^{\infty}\mathop{\rm co}(\xi_{l_{n}})$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and

$F\left(\mathcal{C}+\sum_{n=1}^{\infty}\mathop{\rm
co}(\xi_{l_{n}})\right)\supseteq F(\mathcal{C})$ (3.14)

for every $\mathcal{C}\in\mathcal{A}$.

###### Proof.

By Lemma 3.25 we may assume that the series $\sum_{n=1}^{\infty}\mathop{\rm
co}(\xi_{k_{n}})$ converges u.c.s. in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ for
every increasing sequence of positive integers $(k_{n})_{n\in\mathbb{N}}$.
Moreover, it follows from the nature of the statement of the lemma that we may
restrict to the case when $\mathcal{A}$ is a singleton,
$\mathcal{A}=\\{\mathcal{C}\\}$.

Fix a base point $z\in X$ along with the following objects:

  * •

a sequence $(V_{n})_{n\in\mathbb{N}}$ containing those and only those elements
of a fixed countable basis of $\mathbb{T}^{1}$, which intersect the set
$F(\mathcal{C})$,

  * •

a local base $(U_{n})_{n\in\mathbb{N}}$ at $z$ in $X$ with $U_{n+1}\subseteq
U_{n}$ for every $n\in\mathbb{N}$,

  * •

a local base $(W_{n})_{n\in\mathbb{N}}$ at $1$ in $\mathbb{T}^{1}$ and a
sequence $(\widetilde{V}_{n})_{n\in\mathbb{N}}$ of open subsets of
$\mathbb{T}^{1}$ such that $F(\mathcal{C})\cap\widetilde{V}_{n}\neq\emptyset$,
$W_{n+1}+W_{n+1}\subseteq W_{n}$ and
$\widetilde{V}_{n}+\sum_{i=1}^{n}W_{n}\subseteq V_{n}$ for every
$n\in\mathbb{N}$.

Proceeding by induction on $n$, we shall construct three sequences:

  * •

a sequence $(\widetilde{U}_{n})_{n\in\mathbb{N}}$ of neighbourhoods of $z$ in
$X$,

  * •

a sequence $(\gamma_{n})_{n\in\mathbb{N}}$ of elements of $\Gamma$,

  * •

an increasing sequence $(k_{n})_{n\in\mathbb{N}}$ of positive integers,

such that the following conditions hold for every $n\in\mathbb{N}$:

  1. 1.

$\widetilde{U}_{n+1}\subseteq\widetilde{U}_{n}\subseteq U_{n}$,

  2. 2.

$T_{\gamma_{n}}z\in\widetilde{U}_{n}$ and
$\mathcal{C}(\gamma_{n},z)\in\widetilde{V}_{n}$,

  3. 3.

$\xi_{k_{l}}(\widetilde{U}_{n})-\xi_{k_{l}}(\widetilde{U}_{n})\subseteq W_{n}$
for $1\leq l\leq n-1$,

  4. 4.

$\mathop{\rm
co}(\xi_{k_{n}})(\gamma_{l},z)=\xi_{k_{n}}(T_{\gamma_{l}}z)-\xi_{k_{n}}(z)\in
W_{n+1}$ for $1\leq l\leq n$.

First we define $\widetilde{U}_{1}$, $\gamma_{1}$ and $k_{1}$. Set
$\widetilde{U}_{1}=U_{1}$. Since $\widetilde{V}_{1}$ intersects
$F(\mathcal{C})$, there is $\gamma_{1}\in\Gamma$ with
$T_{\gamma_{1}}z\in\widetilde{U}_{1}$ and
$\mathcal{C}(\gamma_{1},z)\in\widetilde{V}_{1}$. Finally, by our assumptions
on the coboundaries $\mathop{\rm co}(\xi_{i})$, there is $k_{1}\in\mathbb{N}$
such that $\mathop{\rm
co}(\xi_{k_{1}})(\gamma_{1},z)=\xi_{k_{1}}(T_{\gamma_{1}}z)-\xi_{k_{1}}(z)\in
W_{2}$.

Now assume that $\widetilde{U}_{l}$, $\gamma_{l}$ and $k_{l}$ are given for
$l=1,\dots,n$; we shall define $\widetilde{U}_{n+1}$, $\gamma_{n+1}$ and
$k_{n+1}$. First, use continuity of the maps $\xi_{k_{1}},\dots,\xi_{k_{n}}$
to find a neighbourhood $\widetilde{U}_{n+1}\subseteq\widetilde{U}_{n}\cap
U_{n+1}$ of $z$ in $X$ so that
$\xi_{k_{l}}(\widetilde{U}_{n+1})-\xi_{k_{l}}(\widetilde{U}_{n+1})\subseteq
W_{n+1}$ for every $1\leq l\leq n$. Further, since
$F(\mathcal{C})\cap\widetilde{V}_{n+1}\neq\emptyset$, there is
$\gamma_{n+1}\in\Gamma$ with $T_{\gamma_{n+1}}z\in\widetilde{U}_{n+1}$ and
$\mathcal{C}(\gamma_{n+1},z)\in\widetilde{V}_{n+1}$. Finally, by our
assumptions on the coboundaries $\mathop{\rm co}(\xi_{i})$, there is
$k_{n+1}>k_{n}$ such that $\mathop{\rm
co}(\xi_{k_{n+1}})(\gamma_{l},z)=\xi_{k_{n+1}}(T_{\gamma_{l}}z)-\xi_{k_{n+1}}(z)\in
W_{n+2}$ for $1\leq l\leq n+1$. This finishes the induction step of our
construction.

We finish the proof by showing that the sequence $(k_{n})_{n\in\mathbb{N}}$
constructed above satisfies the conclusion of the lemma. First,
$(k_{n})_{n\in\mathbb{N}}$ is increasing by definition. Let
$(l_{n})_{n\in\mathbb{N}}$ be a subsequence of $(k_{n})_{n\in\mathbb{N}}$ and
write $l_{n}=k_{t(n)}$ for $n\in\mathbb{N}$. The series
$\sum_{n=1}^{\infty}\mathop{\rm co}(\xi_{l_{n}})$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ by our argument from the beginning of the
proof; denote the sum of this series by $\mathcal{D}$. To verify (3.14), fix a
neighbourhood $U$ of $z$ in $X$ and an open set $V\subseteq\mathbb{T}^{1}$
intersecting $F(\mathcal{C})$; we need to find $\gamma\in\Gamma$ with
$T_{\gamma}z\in U$ and $(\mathcal{C}+\mathcal{D})(\gamma,z)\in V$. Choose
$n\in\mathbb{N}$ with $\widetilde{U}_{n}\subseteq U$ and
$\overline{V_{n}}\subseteq V$. If $p\geq n$ then, by virtue of (1)–(4) and by
our choice of the sets $V_{i},W_{i},\widetilde{V}_{i}$,

$\begin{split}&\left(\mathcal{C}+\sum_{m=1}^{p}\mathop{\rm
co}(\xi_{l_{m}})\right)(\gamma_{n},z)=\mathcal{C}(\gamma_{n},z)+\sum_{m=1}^{p}\mathop{\rm
co}\left(\xi_{k_{t(m)}}\right)(\gamma_{n},z)\\\
&=\mathcal{C}(\gamma_{n},z)+\sum_{\begin{subarray}{c}1\leq m\leq p\\\ t(m)\leq
n-1\end{subarray}}\left(\xi_{k_{t(m)}}(T_{\gamma_{n}}z)-\xi_{k_{t(m)}}(z)\right)+\sum_{\begin{subarray}{c}1\leq
m\leq p\\\ n\leq
t(m)\end{subarray}}\left(\xi_{k_{t(m)}}(T_{\gamma_{n}}z)-\xi_{k_{t(m)}}(z)\right)\\\
&\in\widetilde{V}_{n}+\sum_{\begin{subarray}{c}1\leq m\leq p\\\ t(m)\leq
n-1\end{subarray}}W_{n}+\sum_{\begin{subarray}{c}1\leq m\leq p\\\ n\leq
t(m)\end{subarray}}W_{t(m)+1}\subseteq\widetilde{V}_{n}+\sum_{i=1}^{n-1}W_{n}+\sum_{i=n+1}^{t(p)+1}W_{i}\\\
&\subseteq\widetilde{V}_{n}+\sum_{i=1}^{n-1}W_{n}+W_{n}=\widetilde{V}_{n}+\sum_{i=1}^{n}W_{n}\subseteq
V_{n}.\end{split}$

Consequently,

$(\mathcal{C}+\mathcal{D})(\gamma_{n},z)=\left(\mathcal{C}+\sum_{m=1}^{\infty}\mathop{\rm
co}(\xi_{l_{m}})\right)(\gamma_{n},z)\in\overline{V_{n}}\subseteq V.$

Since also $T_{\gamma_{n}}z\in\widetilde{U}_{n}\subseteq U$, the proof of the
lemma is finished. ∎

###### Proof of Theorem 3.23.

Before turning to the proof, notice the following facts. First, since the
groups $F(\mathcal{C}_{n})$ ($n\in\mathbb{N}$) are pair-wise group-disjoint,
it follows from Remark 2.5 that

$F(\chi_{1}\mathcal{C}_{n})=\chi_{1}F(\mathcal{C}_{n})\perp\chi_{2}F(\mathcal{C}_{m})=F(\chi_{2}\mathcal{C}_{m})$

for all $\chi_{1},\chi_{2}\in G^{*}$ and $n\neq m$. Further, by virtue of
Lemma 2.4 and Remark 2.5, we may assume that all the groups
$F(\mathcal{C}_{n})$ ($n\in\mathbb{N}$) are totally disconnected and hence
that all the groups $F(\chi\mathcal{C}_{n})=\chi F(\mathcal{C}_{n})$ ($\chi\in
G^{*}$, $n\in\mathbb{N}$) are finite. Finally, by Lemma 3.25, we may assume
that the series $\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ (and hence the series
$\sum_{n=1}^{\infty}\chi\mathcal{C}_{k_{n}}$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ for every $\chi\in G^{*}$) for every
increasing sequence of positive integers $(k_{n})_{n\in\mathbb{N}}$. We divide
the proof into two steps.

_1st step._ Since the group $G$ is second countable, its dual group $G^{*}$ is
countable. Write $G^{*}=\\{\chi_{m}:m\in\mathbb{N}\\}$. We show that there
exists a double sequence of positive integers $(k_{m,n})_{m,n\in\mathbb{N}}$
such that the following conditions hold for every $m\in\mathbb{N}$:

  1. (1)

the sequence $(k_{m,n})_{n\in\mathbb{N}}$ is increasing and contains
$(k_{m+1,n})_{n\in\mathbb{N}}$ as its subsequence,

  2. (2)

either all the groups $F(\chi_{m}\mathcal{C}_{k_{m,n}})$ ($n\in\mathbb{N}$)
are trivial or none of them is trivial,

  3. (3)

if $F(\chi_{m}\mathcal{C}_{k_{m,n}})\neq 1$ for every $n\in\mathbb{N}$ then

$F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)=\mathbb{T}^{1}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{l_{n}})}$

for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of the sequence
$(k_{m,n})_{n\in\mathbb{N}}$,

  4. (4)

if $F(\chi_{m}\mathcal{C}_{k_{m,n}})=1$ for every $n\in\mathbb{N}$ then

$F\left(\sum_{l\in
B}\chi_{m}\mathcal{C}_{l}+\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)\supseteq\sum_{l\in
B}F(\chi_{m}\mathcal{C}_{l})$

for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of the sequence
$(k_{m,n})_{n\geq m+1}$ and every subset $B$ of the set
$\\{1,2,\dots,k_{m,m}-1\\}$.

We shall construct the required double sequence $(k_{m,n})_{m,n\in\mathbb{N}}$
by induction on $m$. So fix $m\in\mathbb{N}$ and assume that
$(k_{j,n})_{n\in\mathbb{N}}$ ($j=1,\dots,m-1$) are sequences of positive
integers satisfying conditions expressed in (1)–(4). In order to construct
$(k_{m,n})_{n\in\mathbb{N}}$, we distinguish two cases.

First assume that $F(\chi_{m}\mathcal{C}_{k_{m-1,n}})\neq 1$ for infinitely
many indexes $n\in\mathbb{N}$. Let $(t_{n})_{n\in\mathbb{N}}$ be a subsequence
of $(k_{m-1,n})_{n\in\mathbb{N}}$ with $F(\chi_{m}\mathcal{C}_{t_{n}})\neq 1$
for every $n\in\mathbb{N}$. Since the groups $F(\chi_{m}\mathcal{C}_{t_{n}})$
($n\in\mathbb{N}$) are pair-wise group-disjoint, Lemma 3.26 yields a
subsequence $(k_{m,n})_{n\in\mathbb{N}}$ of $(t_{n})_{n\in\mathbb{N}}$ such
that

$F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)=\mathbb{T}^{1}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{l_{n}})}$

for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of
$(k_{m,n})_{n\in\mathbb{N}}$. It follows directly from our definition of
$(k_{m,n})_{n\in\mathbb{N}}$ that conditions (1)–(4) hold true.

Now assume that $F(\chi_{m}\mathcal{C}_{k_{m-1,n}})=1$ (that is,
$\chi_{m}\mathcal{C}_{k_{m-1,n}}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$) for all
but finitely many $n\in\mathbb{N}$. Let $(t_{n})_{n\in\mathbb{N}}$ be a
subsequence of $(k_{m-1,n})_{n\in\mathbb{N}}$ with
$\chi_{m}\mathcal{C}_{t_{n}}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ for every
$n\in\mathbb{N}$. Set $k_{m,i}=t_{i}$ for $i=1,\dots,m$ and let $\mathcal{A}$
be the set of all extensions of the form $\sum_{l\in
B}\chi_{m}\mathcal{C}_{l}$, where $B\subseteq\\{1,2,\dots,k_{m,m}-1\\}$. By
virtue of Lemma 3.27, there is a subsequence $(k_{m,n})_{n\geq m+1}$ of
$(t_{n})_{n\geq m+1}$ such that

$F\left(\mathcal{C}+\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)\supseteq
F(\mathcal{C})$

for every subsequence $(l_{n})_{n\in\mathbb{N}}$ of $(k_{m,n})_{n\geq m+1}$
and for every $\mathcal{C}\in\mathcal{A}$. It follows directly from our
definition of $(k_{m,n})_{n\in\mathbb{N}}$ that conditions (1)–(3) hold true.
Condition (4) follows from the definition of $(k_{m,n})_{n\geq m+1}$ and from
the equality $F(\sum_{l\in B}\chi_{m}\mathcal{C}_{l})=\sum_{l\in
B}F(\chi_{m}\mathcal{C}_{l})$ for every subset $B$ of
$\\{1,2,\dots,k_{m,m}-1\\}$, the latter being secured by Proposition 3.19.

_2nd step._ For $n\in\mathbb{N}$ set $k_{n}=k_{n,n}$. Clearly,
$(k_{n})_{n\in\mathbb{N}}$ is an increasing sequence of positive integers. We
show that it satisfies the condition expressed in Theorem 3.23.

The convergence of the series $\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}$ u.c.s.
in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ has already been justified at the
beginning of the proof. In order to verify (3.11), fix a character $\chi$ of
$G$ and choose $m\in\mathbb{N}$ with $\chi=\chi_{m}$. Then
$(k_{n,n})_{n=m}^{\infty}$ is a subsequence of $(k_{m,n})_{n\in\mathbb{N}}$
and hence, by condition (2), there are two possibilities:

  1. (i)

$F(\chi_{m}\mathcal{C}_{k_{n,n}})=1$ for every $n\geq m$,

  2. (ii)

$F(\chi_{m}\mathcal{C}_{k_{n,n}})\neq 1$ for every $n\geq m$.

We wish to show that in both these cases,

$F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{k_{n}}\right)\supseteq\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n}})}.$
(3.15)

First consider case (i). Set $B=\\{k_{n,n}:1\leq n\leq m-1\\}$ and
$(l_{n})_{n\in\mathbb{N}}=(k_{n,n})_{n\geq m+1}$. Clearly, $B$ is a subset of
$\\{1,2,\dots,k_{m,m}-1\\}$. Moreover, since $(l_{n})_{n\in\mathbb{N}}$ is a
subsequence of $(k_{m+1,n})_{n\geq m+1}$ by definition and since
$(k_{m+1,n})_{n\geq m+1}$ is a subsequence of $(k_{m,n})_{n\geq m+1}$, it
follows that $(l_{n})_{n\in\mathbb{N}}$ is a subsequence of $(k_{m,n})_{n\geq
m+1}$. Thus, by applying condition (4) and the inclusion
$\chi_{m}\mathcal{C}_{k_{n,n}}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$ for every
$n\geq m$, we obtain the desired inclusion (3.15):

$\begin{split}F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{k_{n}}\right)&=F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)\\\
&=F\left(\sum_{n=1}^{m-1}\chi_{m}\mathcal{C}_{k_{n,n}}+\chi_{m}\mathcal{C}_{k_{m,m}}+\sum_{n=m+1}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)\\\
&=F\left(\sum_{l\in
B}\chi_{m}\mathcal{C}_{l}+\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)\supseteq\sum_{l\in
B}F(\chi_{m}\mathcal{C}_{l})\\\
&=\overline{\sum_{n=1}^{m-1}F(\chi_{m}\mathcal{C}_{k_{n,n}})}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n,n}})}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n}})}.\end{split}$

Now consider case (ii). Set
$(l_{n})_{n\in\mathbb{N}}=(k_{n,n})_{n=m}^{\infty}$. Since
$(l_{n})_{n\in\mathbb{N}}$ is a subsequence of $(k_{m,n})_{n\in\mathbb{N}}$,
condition (3) yields

$F\left(\sum_{n=m}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)=F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{l_{n}}\right)=\mathbb{T}^{1}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{l_{n}})}=\overline{\sum_{n=m}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n,n}})}.$

Further, since the groups $F(\chi_{m}\mathcal{C}_{k_{n,n}})$ ($1\leq n\leq
m-1$) are pair-wise group-disjoint and finite, it follows from Proposition
3.19 that the group

$F\left(\sum_{n=1}^{m-1}\chi_{m}\mathcal{C}_{k_{n,n}}\right)=\sum_{n=1}^{m-1}F(\chi_{m}\mathcal{C}_{k_{n,n}})$

is also finite. Thus, by virtue of Corollary 3.20,

$\begin{split}F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{k_{n}}\right)&=F\left(\sum_{n=1}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)=F\left(\sum_{n=1}^{m-1}\chi_{m}\mathcal{C}_{k_{n,n}}+\sum_{n=m}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)\\\
&=F\left(\sum_{n=1}^{m-1}\chi_{m}\mathcal{C}_{k_{n,n}}\right)+F\left(\sum_{n=m}^{\infty}\chi_{m}\mathcal{C}_{k_{n,n}}\right)\\\
&=\mathbb{T}^{1}\supseteq\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n,n}})}=\overline{\sum_{n=1}^{\infty}F(\chi_{m}\mathcal{C}_{k_{n}})}.\end{split}$

This verifies the desired inclusion (3.15) also in case (ii).

To summarize, for every $\chi\in G^{*}$, inclusion (3.15) yields

$\begin{split}\chi
F\left(\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}\right)&=F\left(\chi\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}\right)=F\left(\sum_{n=1}^{\infty}\chi\mathcal{C}_{k_{n}}\right)\supseteq\overline{\sum_{n=1}^{\infty}F(\chi\mathcal{C}_{k_{n}})}\\\
&=\overline{\sum_{n=1}^{\infty}\chi
F(\mathcal{C}_{k_{n}})}=\overline{\bigcup_{n\in\mathbb{N}}\sum_{l=1}^{n}\chi
F(\mathcal{C}_{k_{l}})}=\chi\left(\overline{\bigcup_{n\in\mathbb{N}}\sum_{l=1}^{n}F(\mathcal{C}_{k_{l}})}\right)\\\
&=\chi\left(\overline{\sum_{n=1}^{\infty}F(\mathcal{C}_{k_{n}})}\right).\end{split}$

Thus, since $F\left(\sum_{n=1}^{\infty}\mathcal{C}_{k_{n}}\right)$ is a closed
subgroup of $G$, inclusion (3.11) from the theorem follows at last. ∎

######  Proposition 3.28.

Let $\Gamma\in\mathsf{LCGp}\setminus\mathsf{CGp}$ be amenable, $X$ be a
compact space and $G\in\mathsf{CAGp}$ be connected. Assume that
$\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ is a minimal flow with
a free point. If all $\Gamma$, $X$ and $G$ are second countable then there
exists a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ such that the groups $F(\mathcal{C}_{n})$ are pair-wise
group-disjoint, the series $\sum_{n=1}^{\infty}\mathcal{C}_{n}$ converges
u.c.s. in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ and

$F\left(\sum_{n=1}^{\infty}\mathcal{C}_{n}\right)=e\subseteq
G=\overline{\sum_{n=1}^{\infty}F\left(\mathcal{C}_{n}\right)}.$ (3.16)

###### Proof.

Denote by $\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ the closure of
$\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$ in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. By our assumptions, $G$ contains a dense one-parameter
subgroup and so it has a dense identity arc-component $G_{a}$. Consequently,
by virtue of [9, Theorem 6], there is
$\mathcal{C}\in\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ such that the
induced skew product $\mathcal{F}_{\mathcal{C}}$ on $X\times G$ is
topologically transitive (that is, possesses a dense orbit). By our definition
of the functor $F$, $F(\mathcal{C})=G$. Now fix a sequence
$(\xi_{n})_{n\in\mathbb{N}}$ of continuous maps $X\to G$ such that
$\mathop{\rm co}(\xi_{n})\to\mathcal{C}$ u.c.s. in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. Set $\mathcal{C}_{1}=\mathcal{C}$,
$\mathcal{C}_{2}=-\mathop{\rm co}(\xi_{1})=\mathop{\rm
co}\left(-\xi_{1}\right)$ and $\mathcal{C}_{n}=\mathop{\rm
co}(\xi_{n-2})-\mathop{\rm co}(\xi_{n-1})=\mathop{\rm
co}\left(\xi_{n-2}-\xi_{n-1}\right)$ for $n\geq 3$. Then
$\sum_{i=1}^{n}\mathcal{C}_{i}=\mathcal{C}-\mathop{\rm co}(\xi_{n-1})$ for
every $n\geq 2$ and so the series $\sum_{n=1}^{\infty}\mathcal{C}_{n}$
converges u.c.s. to the identity $e$ of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. The two equalities from (3.16) now hold obviously.
Finally, since $F(\mathcal{C}_{1})=G$ and $F(\mathcal{C}_{n})=e$ for every
$n\geq 2$, the groups $F(\mathcal{C}_{n})$ ($n\in\mathbb{N}$) are pair-wise
group-disjoint. ∎

######  Theorem 3.29.

Let $\Gamma\in\mathsf{LCGp}\setminus\mathsf{CGp}$ be amenable, $X$ be a
compact space and $G\in\mathsf{CAGp}$ be connected. Assume that
$\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ is a minimal flow with
a free point $z$. If all $\Gamma$, $X$ and $G$ are second countable then there
exists a sequence $(\xi_{n})_{n\in\mathbb{N}}$ of continuous maps $X\to G$
such that for every increasing sequence of positive integers
$(k_{n})_{n\in\mathbb{N}}$, the series $\sum_{n=1}^{\infty}\mathop{\rm
co}(\xi_{k_{n}})$ converges u.c.s. in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$
and

$F\left(\sum_{n=1}^{\infty}\mathop{\rm co}(\xi_{k_{n}})\right)=G\supseteq
e=\overline{\sum_{n=1}^{\infty}F\left(\mathop{\rm co}(\xi_{k_{n}})\right)}.$

Before turning to the proof of Theorem 3.29, we prove the following auxiliary
lemma.

######  Lemma 3.30.

Under the assumptions of Theorem 3.29, the following statement holds. Given a
neighbourhood $U$ of $z$ in $X$, nonempty open sets
$V_{1},\dots,V_{n}\subseteq G$, a compact set $K\subseteq\Gamma$ and a
neighbourhood $W$ of $e$ in $G$, there exist a continuous map $\xi\colon X\to
G$ and points $\gamma^{1},\dots,\gamma^{n}\in\Gamma$ such that

  1. (i)

$T_{\gamma^{j}}z\in U$ and $\xi(T_{\gamma^{j}}z)-\xi(z)\in V_{j}$ for every
$j=1,\dots,n$ and

  2. (ii)

$\xi(T_{\gamma}x)-\xi(x)\in W$ for all $\gamma\in K$ and $x\in X$.

###### Proof.

Let $\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ be the closure of
$\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$ in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. By our assumptions, $G$ has a dense identity arc-
component $G_{a}$. Consequently, by virtue of [9, Theorem 6], the extensions
$\mathcal{C}\in\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ with
$F(\mathcal{C})=G$ form a residual subset of
$\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ (see also the proof of
Proposition 3.28). In particular, such an extension $\mathcal{C}$ exists
within $[K\times X;W]\cap\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$.
Thus, there are $\mathcal{C}\in\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$
and $\gamma^{1},\dots,\gamma^{n}\in\Gamma$ such that

  * •

$T_{\gamma^{j}}z\in U$ and $\mathcal{C}(\gamma^{j},z)\in V_{j}$ for every
$j=1,\dots,n$ and

  * •

$\mathcal{C}(\gamma,x)\in W$ for all $\gamma\in K$ and $x\in X$.

The two conditions above are clearly open with respect to
$\mathcal{C}\in\mathop{\rm\overline{\bf{B}}_{\mathcal{F}}}(G)$ and hence they
are satisfied also by an appropriate coboundary. That is, there is a
continuous map $\xi\colon X\to G$ such that

  * •

$T_{\gamma^{j}}z\in U$ and $\mathop{\rm co}(\xi)(\gamma^{j},z)\in V_{j}$ for
every $j=1,\dots,n$ and

  * •

$\mathop{\rm co}(\xi)(\gamma,x)\in W$ for all $\gamma\in K$ and $x\in X$.

This verifies both conditions (i) and (ii). ∎

###### Proof of Theorem 3.29.

We start by fixing the following objects:

  * •

a local base $(U_{n})_{n\in\mathbb{N}}$ at $z$ in $X$ with $U_{n+1}\subseteq
U_{n}$ for every $n\in\mathbb{N}$,

  * •

a sequence $(K_{n})_{n\in\mathbb{N}}$ of compact subsets of $\Gamma$, whose
interiors cover $\Gamma$ and which satisfy $K_{n}\subseteq K_{n+1}$ for every
$n\in\mathbb{N}$,

  * •

a basis $(V_{n})_{n\in\mathbb{N}}$ for the topology of $G$,

  * •

a local base $(W_{n})_{n\in\mathbb{N}}$ at $e$ in $G$ and a sequence
$(\widetilde{V}_{n})_{n\in\mathbb{N}}$ of nonempty open subsets of $G$ with
$W_{n+1}+W_{n+1}\subseteq W_{n}$ and $\widetilde{V}_{i}+W_{n}+W_{n}\subseteq
V_{i}$ for $1\leq i\leq n<\infty$.

We claim that there exist

  * •

a sequence $(\xi_{n})_{n\in\mathbb{N}}$ of continuous maps $X\to G$,

  * •

a sequence $(\widetilde{U}_{n})_{n\in\mathbb{N}}$ of neighbourhoods of $z$ in
$X$ and

  * •

a family $\gamma_{n}^{j}$ ($1\leq j\leq n<\infty$) of elements of $\Gamma$,

such that the following conditions hold:

  1. (a)

$\widetilde{U}_{n}\subseteq U_{n}$ for every $n\in\mathbb{N}$,

  2. (b)

$\xi_{i}(x)-\xi_{i}(z)\in W_{2n-1}$ for $x\in\widetilde{U}_{n}$ and $1\leq
i\leq n-1$,

  3. (c)

$T_{\gamma_{n}^{j}}z\in\widetilde{U}_{n}$ and
$\xi_{n}(T_{\gamma_{n}^{j}}z)-\xi_{n}(z)\in\widetilde{V}_{j}$ for $1\leq j\leq
n<\infty$,

  4. (d)

$\xi_{n}(T_{\gamma}x)-\xi_{n}(x)\in W_{n}$ for $\gamma\in
K_{n}\cup\\{\gamma_{k}^{j}:1\leq j\leq k\leq n-1\\}$ and $x\in X$.

The construction of these objects follows easily by induction on $n$. Indeed,
assume that the maps $\xi_{1},\dots,\xi_{n-1}$, the sets
$\widetilde{U}_{1},\dots,\widetilde{U}_{n-1}$ and the points $\gamma_{k}^{j}$
($1\leq j\leq k\leq n-1$) are given. Fix a neighbourhood $\widetilde{U}_{n}$
of $z$ in $X$ small enough so that $\widetilde{U}_{n}\subseteq U_{n}$ and
$\xi_{i}(x)-\xi_{i}(z)\in W_{2n-1}$ for all $x\in\widetilde{U}_{n}$ and
$i=1,\dots,n-1$. Finally, use Lemma 3.30 to find a continuous map
$\xi_{n}\colon X\to G$ and points
$\gamma_{n}^{1},\dots,\gamma_{n}^{n}\in\Gamma$ such that

  * •

$T_{\gamma_{n}^{j}}z\in\widetilde{U}_{n}$ and
$\xi_{n}(T_{\gamma_{n}^{j}}z)-\xi_{n}(z)\in\widetilde{V}_{j}$ for
$j=1,\dots,n$ and

  * •

$\xi_{n}(T_{\gamma}x)-\xi_{n}(x)\in W_{n}$ for all $\gamma\in
K_{n}\cup\\{\gamma_{k}^{j}:1\leq j\leq k\leq n-1\\}$ and $x\in X$.

Then, clearly, all the conditions (a)–(d) hold true.

We show that the sequence $(\xi_{n})_{n\in\mathbb{N}}$ satisfies the
conclusion of the theorem. To this end, fix an increasing sequence of positive
integers $(k_{n})_{n\in\mathbb{N}}$. First, we verify the u.c.s. convergence
of $\sum_{n=1}^{\infty}\mathop{\rm co}(\xi_{k_{n}})$ in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. So let $K\subseteq\Gamma$ be a compact set and $W$ be a
neighbourhood of $e$ in $G$. Choose $n_{0}\in\mathbb{N}$ with $K\subseteq
K_{n_{0}}$ and $W_{n_{0}}\subseteq W$. Let $\gamma\in K$, $x\in X$, $n\geq
n_{0}$ and $p\in\mathbb{N}$. Since $K_{k_{l}}\supseteq K_{l}\supseteq
K_{n_{0}}\supseteq K$ for every $l\geq n_{0}$, it follows from condition (d)
above that

$\begin{split}\sum_{l=1}^{n+p}\mathop{\rm
co}(\xi_{k_{l}})(\gamma,x)-\sum_{l=1}^{n}\mathop{\rm
co}(\xi_{k_{l}})(\gamma,x)&=\sum_{l=n+1}^{n+p}\mathop{\rm
co}(\xi_{k_{l}})(\gamma,x)\\\
&=\sum_{l=n+1}^{n+p}\left(\xi_{k_{l}}(T_{\gamma}x)-\xi_{k_{l}}(x)\right)\\\
&\in\sum_{l=n+1}^{n+p}W_{k_{l}}\subseteq\sum_{l=n+1}^{n+p}W_{l}\subseteq
W_{n}\subseteq W_{n_{0}}\subseteq W.\end{split}$

By completeness of $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, the series
$\sum_{n=1}^{\infty}\mathop{\rm co}(\xi_{k_{n}})$ converges u.c.s. in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Denote the sum of this series by
$\mathcal{C}$.

We show that $\mathcal{C}(\gamma_{k_{n}}^{n},z)\in\overline{V_{n}}$ for every
$n\in\mathbb{N}$. So fix $n\in\mathbb{N}$ and let $p\in\mathbb{N}$. We shall
make use of the following observations.

  * •

Given $1\leq l\leq n-1$, we have
$T_{\gamma_{k_{n}}^{n}}z\in\widetilde{U}_{k_{n}}$ by virtue of (c). Hence
$\xi_{k_{l}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{l}}(z)\in W_{2k_{n}-1}\subseteq
W_{2n-1}$ by virtue of (b).

  * •

We have
$\xi_{k_{n}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{n}}(z)\in\widetilde{V}_{n}$ by
virtue of (c).

  * •

Given $n+1\leq l\leq n+p$, we have
$\xi_{k_{l}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{l}}(z)\in W_{k_{l}}\subseteq
W_{l}$ by virtue of (d).

These observations yield

$\begin{split}\sum_{l=1}^{n+p}\mathop{\rm
co}(\xi_{k_{l}})(\gamma_{k_{n}}^{n},z)&=\sum_{l=1}^{n-1}\left(\xi_{k_{l}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{l}}(z)\right)+\left(\xi_{k_{n}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{n}}(z)\right)+\\\
&+\sum_{l=n+1}^{n+p}\left(\xi_{k_{l}}(T_{\gamma_{k_{n}}^{n}}z)-\xi_{k_{l}}(z)\right)\\\
&\in\sum_{l=1}^{n-1}W_{2n-1}+\widetilde{V}_{n}+\sum_{l=n+1}^{n+p}W_{l}\subseteq\sum_{l=1}^{n-1}W_{n+l}+\widetilde{V}_{n}+\sum_{l=n+1}^{n+p}W_{l}\\\
&\subseteq W_{n}+\widetilde{V}_{n}+W_{n}\subseteq V_{n}.\end{split}$

Since these inclusions hold for every $p\in\mathbb{N}$, it follows that,
indeed, $\mathcal{C}(\gamma_{k_{n}}^{n},z)\in\overline{V_{n}}$.

We finish the proof of the theorem by showing that $F(\mathcal{C})=G$. To this
end, fix a neighbourhood $U$ of $z$ in $X$ and a nonempty open set $V\subseteq
G$; we need to find $\gamma\in\Gamma$ with $T_{\gamma}z\in U$ and
$\mathcal{C}(\gamma,z)\in V$. Choose $n\in\mathbb{N}$ with $U_{n}\subseteq U$
and $\overline{V_{n}}\subseteq V$, and set $\gamma=\gamma_{k_{n}}^{n}$. Then
$T_{\gamma}z\in\widetilde{U}_{k_{n}}\subseteq U_{k_{n}}\subseteq
U_{n}\subseteq U$ by virtue of (c) and (a), and
$\mathcal{C}(\gamma,z)\in\overline{V_{n}}\subseteq V$ by our argument above. ∎

###  3.4 The ext-topology

In the previous section we had an opportunity to recall [9, Theorem 6], which
states that under mild assumptions on a minimal flow $\mathcal{F}$ and a group
$G\in\mathsf{CAGp}$, a generic extension from the closure of $\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)$ in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is minimal; in
particular, $\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$ is not a closed subgroup of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ in the topology of u.c.s. convergence.
Consequently, the quotient topology on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$
is not Hausdorff and so $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ with this
topology is not a topological group in the sense used in this work.

Our aim in this section is to introduce a natural (Hausdorff) group topology
$\tau_{ext}$ on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$. We are motivated by
our results from the preceding sections, where we discussed two continuity
properties of the functor $F$. Being inspired by these considerations, we are
searching for a group topology on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, with
respect to which the map $F\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to
2^{G}$ would be continuous. Our main result in this section, namely Theorem
3.34, shows that the coarsest topology with these properties does exist and is
induced by a natural $2^{G}$-valued translation-invariant metric on
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$. We call this topology the _ext-
topology_ and denote it by $\tau_{ext}$.

Before turning to our results recall that for $G\in\mathsf{CAGp}$, the hyper-
semigroup $2^{G}$ over $G$ carries the Vietoris topology. Moreover, by Lemma
2.2, for every $K\in 2^{G}$, the sets

$\mathcal{U}_{V}(K)=\left\\{H\in 2^{G}:H\subseteq VK\hskip
5.69054pt\text{and}\hskip 5.69054ptK\subseteq VH\right\\}$

form a local base at $K$ in $2^{G}$, when $V$ runs through the neighbourhoods
of $e$ in $G$. Consequently, given a net $(H_{i})_{i\in I}$ of closed
subgroups of $G$, the following conditions are equivalent:

  * •

$H_{i}\to e$ in $2^{G}$,

  * •

for every neighbourhood $V$ of $e$ in $G$, there is $i_{0}\in I$ such that
$H_{i}\subseteq V$ for every $i\geq i_{0}$.

######  Lemma 3.31.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $G\in\mathsf{CAGp}$. If $\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ then the following statements hold:

  1. (a)

$F(\mathcal{C})=e$ if and only if $\mathcal{C}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)$,

  2. (b)

$F(\mathcal{C}^{-1})=F(\mathcal{C})^{-1}=F(\mathcal{C})$,

  3. (c)

$F(\mathcal{C}\mathcal{D})\subseteq F(\mathcal{C})F(\mathcal{D})$.

###### Proof.

Statement (a) follows from Theorem 3.7(2). Since the inversion in $G$ is an
automorphism of $G$ and $F(\mathcal{C})$ is a subgroup of $G$, statement (b)
follows from Theorem 3.7(3). To verify statement (c), let $\varphi\colon
G\times G\to G$ be the product in $G$ and for $i=1,2$, let $\mathop{\rm
pr}_{i}\colon G\times G\to G$ be the projection on the $i$-th coordinate. By
Theorem 3.7(3), $\mathop{\rm pr}_{1}F(\mathcal{C},\mathcal{D})=F(\mathop{\rm
pr}_{1}(\mathcal{C},\mathcal{D}))=F(\mathcal{C})$ and, similarly, $\mathop{\rm
pr}_{2}F(\mathcal{C},\mathcal{D})=F(\mathop{\rm
pr}_{2}(\mathcal{C},\mathcal{D}))=F(\mathcal{D})$. Consequently,
$F(\mathcal{C},\mathcal{D})\subseteq F(\mathcal{C})\times F(\mathcal{D})$. By
applying Theorem 3.7(3) once more, we obtain

$F(\mathcal{C}\mathcal{D})=F(\varphi(\mathcal{C},\mathcal{D}))=\varphi(F(\mathcal{C},\mathcal{D}))\subseteq\varphi(F(\mathcal{C})\times
F(\mathcal{D}))=F(\mathcal{C})F(\mathcal{D}),$

as was to be shown. ∎

######  Proposition 3.32.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $G\in\mathsf{CAGp}$. For $\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ set
$\delta(\mathcal{C},\mathcal{D})=F(\mathcal{C}\mathcal{D}^{-1})$. If
$\mathcal{B},\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$
then the following statements hold:

  1. (i)

$\delta(\mathcal{C},\mathcal{D})=e$ if and only if
$\mathcal{C}\simeq\mathcal{D}$,

  2. (ii)

$\delta(\mathcal{C},\mathcal{D})=\delta(\mathcal{D},\mathcal{C})$,

  3. (iii)

$\delta(\mathcal{C},\mathcal{D})\subseteq\delta(\mathcal{C},\mathcal{B})\delta(\mathcal{B},\mathcal{D})$,

  4. (iv)

$\delta(\mathcal{C}\mathcal{B},\mathcal{D}\mathcal{B})=\delta(\mathcal{C},\mathcal{D})$.

######  Remark 3.33.

The proposition states that $\delta$ is a translation-invariant $2^{G}$-valued
pseudo-metric on $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, which identifies
precisely the pairs of equivalent extensions. It follows immediately from the
definition of $\delta$ and from conditions (i)–(iv) above that $\delta$ gives
rise to a translation-invariant $2^{G}$-valued metric on $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)=\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)/\mathop{\rm{\bf
B}_{\mathcal{F}}}(G)=\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)/\simeq$.

###### Proof of Proposition 3.32.

Statement (i) follows from statement (a) in Lemma 3.31. Statement (ii) is a
direct consequence of statement (b) from Lemma 3.31. Statement (iii) follows
from statement (c) of Lemma 3.31:

$\begin{split}\delta(\mathcal{C},\mathcal{D})&=F(\mathcal{C}\mathcal{D}^{-1})=F\left(\left(\mathcal{C}\mathcal{B}^{-1}\right)\left(\mathcal{B}\mathcal{D}^{-1}\right)\right)\subseteq
F\left(\mathcal{C}\mathcal{B}^{-1}\right)F\left(\mathcal{B}\mathcal{D}^{-1}\right)\\\
&=\delta(\mathcal{C},\mathcal{B})\delta(\mathcal{B},\mathcal{D}).\end{split}$

Finally, statement (iv) follows at once from the definition of $\delta$. ∎

Now let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal
flow and $G\in\mathsf{CAGp}$. We wish to introduce a topology on
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ by using the metric $\delta$ defined
above. Given a neighbourhood $V$ of $e$ in $G$ and an extension
$\mathcal{C}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, let $B(\mathcal{C},V)$
be the set of all $\mathcal{D}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ such
that $\delta(\mathcal{D},\mathcal{C})\subseteq V$. As follows from Theorem
3.34 below, the sets $B(\mathcal{C},V)$ form a basis for a group topology on
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$. We shall call it the _ext-topology_
and denote it by $\tau_{ext}$.

######  Theorem 3.34.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $G\in\mathsf{CAGp}$. Then the collection of sets $B(\mathcal{C},V)$, where
$\mathcal{C}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ and $e\in V\subseteq G$
is open, forms a basis for a topology $\tau_{ext}$ on $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$. Moreover, $\tau_{ext}$ is the coarsest of all the
topologies on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, with respect to which
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is a (Hausdorff) topological group and
the mapping $F\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to 2^{G}$ is
continuous.

######  Remark 3.35.

The following facts will follow from our arguments in the proof of the
theorem.

  * •

The topology $\tau_{ext}$ is in fact the coarsest of all the (Hausdorff)
topologies on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, with respect to which
the mapping $F\colon\mathop{\rm{\bf H}_{\mathcal{F}}}(G)\to 2^{G}$ is
continuous and $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is a semi-topological
semigroup (that is, the product in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is
separately continuous).

  * •

For a fixed $\mathcal{C}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, the sets
$B(\mathcal{C},V)$ form a local base at $\mathcal{C}$ in $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ with $\tau_{ext}$, when $V$ runs through the
neighbourhoods of $e$ in $G$.

The following observations will also be useful to us.

  * •

A net $(\mathcal{C}_{i})$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ converges
to $\mathcal{C}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ with respect to
$\tau_{ext}$ if and only if the net $(F(\mathcal{C}_{i}\mathcal{C}^{-1}))$
converges to $e$ in $2^{G}$; this follows from the previous statement.

  * •

If $G$ is a Lie group (that is, if $G_{0}$ is a finite-dimensional torus and
is open in $G$) then $G$ has no small subgroups and the topology $\tau_{ext}$
on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is therefore discrete. Thus, in
particular, the group $\mathop{\rm{\bf H}_{\mathcal{F}}}$ is always discrete.

###### Proof of Theorem 3.34.

First, we verify that the sets $B(\mathcal{C},V)$ form a basis for a Hausdorff
topology on $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$. Their union is clearly the
whole set $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$, since $B(\mathcal{C},V)$
contains $\mathcal{C}$ for all $\mathcal{C}$ and $V$.

Now let $\mathcal{C},\mathcal{D},\mathcal{E}\in\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ and $V,W$ be neighbourhoods of $e$ in $G$ with
$\mathcal{E}\in B(\mathcal{C},V)\cap B(\mathcal{D},W)$; we claim that
$B(\mathcal{E},N)\subseteq B(\mathcal{C},V)\cap B(\mathcal{D},W)$ for an
appropriate neighbourhood $N$ of $e$ in $G$. (In particular, this will verify
the second and hence also the third statement from Remark 3.35.) By
definition, $\delta(\mathcal{E},\mathcal{C})$ and
$\delta(\mathcal{E},\mathcal{D})$ are closed subgroups of $G$ and they are
contained in $V$ and $W$, respectively. Consequently, there is a neighbourhood
$N$ of $e$ in $G$ with $N\delta(\mathcal{E},\mathcal{C})\subseteq V$ and
$N\delta(\mathcal{E},\mathcal{D})\subseteq W$. We claim that
$B(\mathcal{E},N)\subseteq B(\mathcal{C},V)\cap B(\mathcal{D},W)$. Indeed, if
$\mathcal{B}\in B(\mathcal{E},N)$ then, by Proposition 3.32(iii),
$\delta(\mathcal{B},\mathcal{C})\subseteq\delta(\mathcal{B},\mathcal{E})\delta(\mathcal{E},\mathcal{C})\subseteq
N\delta(\mathcal{E},\mathcal{C})\subseteq V$ and hence $\mathcal{B}\in
B(\mathcal{C},V)$. By a similar argument, $B(\mathcal{E},N)\subseteq
B(\mathcal{D},W)$.

We show that $\tau_{ext}$ satisfies the Hausdorff condition. If
$\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ are distinct
then $\delta(\mathcal{C},\mathcal{D})\neq e$ by Proposition 3.32(i). It
follows that $\delta(\mathcal{C},\mathcal{D})\not\subseteq V^{2}$ for an
appropriate neighbourhood $V$ of $e$ in $G$. We claim that
$B(\mathcal{C},V)\cap B(\mathcal{D},V)=\emptyset$. Indeed, if $\mathcal{E}\in
B(\mathcal{C},V)\cap B(\mathcal{D},V)$ then, by Proposition 3.32(iii, ii),

$\delta(\mathcal{C},\mathcal{D})\subseteq\delta(\mathcal{C},\mathcal{E})\delta(\mathcal{E},\mathcal{D})=\delta(\mathcal{E},\mathcal{C})\delta(\mathcal{E},\mathcal{D})\subseteq
VV=V^{2},$

a contradiction.

Now we verify the second statement of the theorem. We start by showing that
$\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ is a topological group when equipped
with the topology $\tau_{ext}$ and that the mapping $F$ is continuous. First,
we verify the continuity of the product in $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$. To this end, fix
$\mathcal{C},\mathcal{D}\in\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ and a
neighbourhood $V$ of $e$ in $G$; we shall find a neighbourhood $W$ of $e$ in
$G$ with $B(\mathcal{C},W)B(\mathcal{D},W)\subseteq
B(\mathcal{C}\mathcal{D},V)$. Choose a neighbourhood $W$ of $e$ in $G$ with
$W^{2}\subseteq V$. Then for all $\mathcal{C}^{\prime}\in B(\mathcal{C},W)$
and $\mathcal{D}^{\prime}\in B(\mathcal{D},W)$,

$\delta(\mathcal{C}^{\prime}\mathcal{D}^{\prime},\mathcal{C}\mathcal{D})\subseteq\delta(\mathcal{C}^{\prime}\mathcal{D}^{\prime},\mathcal{C}^{\prime}\mathcal{D})\delta(\mathcal{C}^{\prime}\mathcal{D},\mathcal{C}\mathcal{D})=\delta(\mathcal{D}^{\prime},\mathcal{D})\delta(\mathcal{C}^{\prime},\mathcal{C})\subseteq
WW\subseteq V$

by virtue of Proposition 3.32. Consequently,
$B(\mathcal{C},W)B(\mathcal{D},W)\subseteq B(\mathcal{C}\mathcal{D},V)$, which
verifies the continuity of the product in $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$. The continuity of the inversion in $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ follows from the identity
$B(\mathcal{C},V)^{-1}=B(\mathcal{C}^{-1},V)$, which holds by Lemma 3.31(b).

To show that $F$ is continuous, fix $\mathcal{C}\in\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$ and a neighbourhood $V$ of $e$ in $G$. By virtue of
Lemma 2.2, it suffices to show that
$F(\mathcal{D})\in\mathcal{U}_{V}(F(\mathcal{C}))$ for every $\mathcal{D}\in
B(\mathcal{C},V)$. If $\mathcal{D}\in B(\mathcal{C},V)$ then

$F(\mathcal{D})=F((\mathcal{D}\mathcal{C}^{-1})\mathcal{C})\subseteq
F(\mathcal{D}\mathcal{C}^{-1})F(\mathcal{C})=\delta(\mathcal{D},\mathcal{C})F(\mathcal{C})\subseteq
VF(\mathcal{C})$

by Lemma 3.31(c), and $F(\mathcal{C})\subseteq VF(\mathcal{D})$ by a similar
argument. Thus, $F(\mathcal{D})\in\mathcal{U}_{V}(F(\mathcal{C}))$ for every
$\mathcal{D}\in B(\mathcal{C},V)$, as was to be shown.

Finally, let $\tau$ be a group topology on $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$, with respect to which the map $F$ is continuous; we
show that $\tau_{ext}$ is coarser than $\tau$. To this end, fix a net
$(\mathcal{C}_{i})$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ and assume that
$\mathcal{C}_{i}\to\mathcal{C}$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ with
respect to $\tau$. Since $\tau$ is a group topology on $\mathop{\rm{\bf
H}_{\mathcal{F}}}(G)$, it follows that $\mathcal{C}_{i}\mathcal{C}^{-1}\to e$
in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ with respect to $\tau$.
Consequently, by continuity of $F$, $F(\mathcal{C}_{i}\mathcal{C}^{-1})\to
F(e)=e$ in $2^{G}$. By the third part of Remark 3.35, it follows that
$\mathcal{C}_{i}\to\mathcal{C}$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(G)$ with
respect to $\tau_{ext}$. This shows that $\tau_{ext}$ is coarser than $\tau$,
as was to be shown. ∎

###  3.5 Lifts of extensions

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
In this section we study the problem of lifting extensions of $\mathcal{F}$
across covering morphisms of abelian topological groups. In Theorem 3.38 we
show that this is always possible if the acting group $\Gamma$ of
$\mathcal{F}$ is simply connected. (On the other hand, it is no longer
possible if the assumption of simple connectedness is moved from $\Gamma$ to
the phase space $X$ of $\mathcal{F}$; this is shown in Example 3.40.) Then we
use this result to find the structure of a real separable Fréchet or Banach
space in the groups $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ for connected
second countable groups $G\in\mathsf{CAGp}$, see Corollary 3.41. Finally, in
Theorem 3.43 we show that for flows $\mathcal{F}$ with simply connected acting
groups $\Gamma\in\mathsf{CLAC}$ and with compact second countable phase spaces
$X$ satisfying $\pi^{1}(X)\neq 0$, the groupoid $\mathop{\rm{\bf
MinH}_{\mathcal{F}}}$ contains an isomorphic copy of the group $\mathbb{R}$.

######  Lemma 3.36.

Let $\mathcal{F}\colon\Gamma\curvearrowright X$ be a minimal flow with a
connected acting group $\Gamma$, $G,G^{\prime}$ be abelian topological groups
and $p\in\mathop{\rm Hom}(G^{\prime},G)$ be a morphism with a totally
disconnected kernel. Given $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ and a continuous map
$\mathcal{C}^{\prime}\colon\Gamma\times X\to G^{\prime}$ with
$p\mathcal{C}^{\prime}=\mathcal{C}$, the following conditions are equivalent:

  1. (a)

$\mathcal{C}^{\prime}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G^{\prime})$,

  2. (b)

$\mathcal{C}^{\prime}(1,x)=e$ for every $x\in X$,

  3. (c)

$\mathcal{C}^{\prime}(1,z)=e$ for some $z\in X$.

######  Remark 3.37.

Our assumptions on $p$ are satisfied if $p$ is a covering morphism. Indeed,
recall that an epimorphism $p\in\mathop{\rm Hom}(G^{\prime},G)$ between
topological groups $G^{\prime},G$ is a covering map if and only if it is a
quotient (equivalently, open) map with a discrete kernel $\ker(p)\subseteq
G^{\prime}$.

###### Proof of Lemma 3.36.

Implication (a)$\Rightarrow$(b) follows from the cocycle identity and
implication (b)$\Rightarrow$(c) is clear. We show that (c)$\Rightarrow$(b). To
this end, consider the map $\varphi\colon\Gamma\to G^{\prime}$, given by
$\varphi(\gamma)=\mathcal{C}^{\prime}(1,T_{\gamma}z)$. Then
$p\varphi(\gamma)=p\mathcal{C}^{\prime}(1,T_{\gamma}z)=\mathcal{C}(1,T_{\gamma}z)=e$
for every $\gamma\in\Gamma$ and so $\varphi$ takes its values in $\ker(p)$.
Since $\varphi$ is continuous, $\Gamma$ is connected and $\ker(p)$ is totally
disconnected, it follows that $\varphi$ is constant with the value
$\varphi(1)=\mathcal{C}^{\prime}(1,z)=e$. That is to say,
$\mathcal{C}^{\prime}(1,x)=e$ for every $x$ from the $\mathcal{F}$-orbit of
$z$. Condition (b) now follows by minimality of $\mathcal{F}$.

We finish the proof by showing that (b)$\Rightarrow$(a). To this end, fix
$x\in X$ and consider the map $\psi\colon\Gamma\times\Gamma\to G^{\prime}$,
given by
$\psi(\alpha,\beta)=\mathcal{C}^{\prime}(\alpha,T_{\beta}x)\mathcal{C}^{\prime}(\beta,x)\mathcal{C}^{\prime}(\alpha\beta,x)^{-1}$.
Then
$p\psi(\alpha,\beta)=\mathcal{C}(\alpha,T_{\beta}x)\mathcal{C}(\beta,x)\mathcal{C}(\alpha\beta,x)^{-1}=e$
for all $\alpha,\beta\in\Gamma$ and so $\psi$ takes its values in $\ker(p)$.
As above, it follows that $\psi$ is constant with the value
$\psi(1,1)=\mathcal{C}^{\prime}(1,x)=e$ and so $\mathcal{C}^{\prime}$
satisfies the cocycle identity at $x$. Since $x\in X$ was arbitrary, this
verifies condition (a). ∎

######  Theorem 3.38.

Let $\mathcal{F}\colon\Gamma\curvearrowright X$ be a minimal flow with
$\Gamma\in\mathsf{CLAC}$ simply connected, $G,G^{\prime}$ be abelian
topological groups and $p\in\mathop{\rm Hom}(G^{\prime},G)$ be a covering
morphism. Then for every $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$
there is a unique $\mathcal{C}^{\prime}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G^{\prime})$ with $p\mathcal{C}^{\prime}=\mathcal{C}$.

######  Remark 3.39.

We wish to add the following observations.

  * •

The theorem applies to the following particular situations:

    1. (a)

$G=G^{\prime}=\mathbb{T}^{1}$ and $p=\kappa_{d}$ ($d\in\mathbb{N}$); this
shows that under the assumptions of the theorem, the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is divisible,

    2. (b)

$G=\mathbb{T}^{1}$, $G^{\prime}=\mathbb{R}$ and
$p\colon\mathbb{R}\to\mathbb{T}^{1}$ is the usual covering morphism.

  * •

If the groups $G,G^{\prime}$ are compact then $p\in\mathop{\rm
Hom}(G^{\prime},G)$ is a covering morphism if and only if it is an epimorphism
with a finite kernel. In Theorem 4.14 we shall see that for compact groups
$G,G^{\prime}$, the extensions $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ lift uniquely across $p$ also when $p\colon
G^{\prime}\to G$ is an epimorphism with a totally disconnected kernel.

  * •

It is a frequent assumption in texts dealing with covering maps that the
domain of a lifted map is an element of $\mathsf{CLAC}$. A special form of our
lifted map $\mathcal{C}$ will allow us to abandon this assumption. That is, we
will not have to assume that $X\in\mathsf{CLAC}$. As a matter of fact, we
shall need to apply Theorem 3.38 several times to a situation when $X$ is a
general, not necessarily locally connected continuum.

###### Proof of Theorem 3.38.

By virtue of Lemma 3.36, we need to lift $\mathcal{C}$ across $p$ to a
continuous map $\mathcal{C}^{\prime}\colon\Gamma\times X\to G^{\prime}$ with
$\mathcal{C}^{\prime}(1,x)=e$ for every $x\in X$. We proceed in four steps.

_1st step._ We define the required lift $\mathcal{C}^{\prime}$ of
$\mathcal{C}$.

Fix $\gamma\in\Gamma$, $x\in X$ and choose a path $f$ in $\Gamma$ from $1$ to
$\gamma$. Then $\widetilde{f}\colon[0,1]\ni t\mapsto\mathcal{C}(f(t),x)\in G$
is a path in $G$ from $e$ to $\mathcal{C}(\gamma,x)$. Since $p$ is a covering
morphism, $\widetilde{f}$ lifts across $p$ to a unique path $\widehat{f}$ in
$G^{\prime}$ starting at $e$. Since $\Gamma$ is simply connected, the endpoint
of $\widehat{f}$ does not depend on $f$, only on $\gamma$ and $x$; we shall
denote this endpoint by $\mathcal{C}^{\prime}(\gamma,x)$. Clearly, the map
$\mathcal{C}^{\prime}\colon\Gamma\times X\to G^{\prime}$ thus defined projects
to $\mathcal{C}$ via $p$ and $\mathcal{C}^{\prime}(1,x)=e$ for every $x\in X$.
We thus need to show that $\mathcal{C}^{\prime}$ is continuous.

_2nd step._ We show that $\mathcal{C}^{\prime}$ is continuous in the second
variable.

Fix $\gamma\in\Gamma$ and $x\in X$; we show that $\mathcal{C}^{\prime}$ is
continuous at $(\gamma,x)$ in the second variable. So let $V^{\prime}$ be a
neighbourhood of $\mathcal{C}^{\prime}(\gamma,x)$ in $G^{\prime}$. Fix a path
$f$ in $\Gamma$ from $1$ to $\gamma$. From our construction of
$\mathcal{C}^{\prime}$ it follows that
$\widehat{f}(t)=\mathcal{C}^{\prime}(f(t),x)$ for every $t\in[0,1]$. Thus, by
continuity of $\widehat{f}$, there exist

  * •

a partition $0=t_{0}<t_{1}<\dots<t_{n}=1$ of $[0,1]$,

  * •

open subsets $V_{1},\dots,V_{n}$ of $G$, each of them evenly covered by $p$

  * •

open subsets $V_{1}^{\prime},\dots,V_{n}^{\prime}$ of $G^{\prime}$,
$V_{i}^{\prime}$ being a $p$-slice over $V_{i}$ for every $i=1,\dots,n$, and
$V_{n}^{\prime}\subseteq V^{\prime}$,

such that

  1. ($*$)

$\mathcal{C}^{\prime}(f([t_{i-1},t_{i}])\times x)\subseteq V_{i}^{\prime}$ for
$i=1,\dots,n$.

By virtue of ($*$),

$\mathcal{C}(f([t_{i-1},t_{i}])\times
x)=p\mathcal{C}^{\prime}(f([t_{i-1},t_{i}])\times x)\subseteq
p(V_{i}^{\prime})=V_{i}$

for $i=1,\dots,n$ and
$\mathcal{C}(f(t_{i}),x)=p\mathcal{C}^{\prime}(f(t_{i}),x)\in
p(V_{i}^{\prime}\cap V_{i+1}^{\prime})$ for $i=1,\dots,n-1$. Consequently, by
continuity of $\mathcal{C}$, there is a neighbourhood $U$ of $x$ in $X$ such
that

  1. (a)

$\mathcal{C}(f([t_{i-1},t_{i}])\times U)\subseteq V_{i}$ for $i=1,\dots,n$,
and

  2. (b)

$\mathcal{C}(f(t_{i})\times U)\subseteq p(V_{i}^{\prime}\cap
V_{i+1}^{\prime})$ for $i=1,\dots,n-1$.

Now let $y\in U$; we show that $\mathcal{C}^{\prime}(\gamma,y)\in V^{\prime}$.
Consider the path $\widetilde{g}\colon[0,1]\ni t\mapsto\mathcal{C}(f(t),y)\in
G$ in $G$ from $e$ to $\mathcal{C}(\gamma,y)$. For $i=1,\dots,n$, let
$\widetilde{g}_{i}$ be the restriction of $\widetilde{g}$ onto
$[t_{i-1},t_{i}]$, $p_{i}\colon V_{i}^{\prime}\to V_{i}$ be the homeomorphism
obtained by restricting $p$ and $\widehat{g}_{i}=p_{i}^{-1}\widetilde{g}_{i}$.
By virtue of (a), all the maps $\widehat{g}_{i}$ are well defined and
continuous. By virtue of (b),

$\widehat{g}_{i}(t_{i})=p_{i}^{-1}\widetilde{g}_{i}(t_{i})=p_{i}^{-1}\mathcal{C}(f(t_{i}),y)=p_{i+1}^{-1}\mathcal{C}(f(t_{i}),y)=p_{i+1}^{-1}\widetilde{g}_{i+1}(t_{i})=\widehat{g}_{i+1}(t_{i})$

for every $i=1,\dots,n-1$. Consequently, the maps $\widehat{g}_{i}$ are
restrictions of a single path $\widehat{g}\colon[0,1]\to G^{\prime}$ in
$G^{\prime}$, which projects to $\widetilde{g}$ via $p$. By our definition of
$\mathcal{C}^{\prime}$, it follows that

$\mathcal{C}^{\prime}(\gamma,y)=\widehat{g}(1)=\widehat{g}_{n}(1)=p_{n}^{-1}\widetilde{g}_{n}(1)\in
V_{n}^{\prime}\subseteq V^{\prime},$

as was to be shown. This shows that $\mathcal{C}^{\prime}$ is continuous in
the second variable.

_3rd step._ We show that $\mathcal{C}^{\prime}$ is (jointly) continuous.

Let $\gamma$, $x$ and $V^{\prime}$ be as in the second step of the proof and
set $V=p(V^{\prime})$; we shall find a neighbourhood $W$ of $\gamma$ in
$\Gamma$ and a neighbourhood $U$ of $x$ in $X$ with
$\mathcal{C}^{\prime}(W\times U)\subseteq V^{\prime}$. Without loss of
generality, we may assume that $V$ is evenly covered by $p$ and $V^{\prime}$
is a $p$-slice over $V$. Since $p$ is an open map by the assumptions, $V$ is a
neighbourhood of $\mathcal{C}(\gamma,x)$ in $G$. Fix an arc-wise connected
neighbourhood $W$ of $\gamma$ in $\Gamma$ and a neighbourhood $U$ of $x$ in
$X$ with $\mathcal{C}(W\times U)\subseteq V$. Since $\mathcal{C}^{\prime}$ is
continuous in the second variable by the second step of the proof, we may
assume that $\mathcal{C}^{\prime}(\gamma\times U)\subseteq V^{\prime}$.

Fix $\delta\in W$ and $y\in U$. Choose a path $h$ in $\Gamma$ with

  * •

$h(0)=1$, $h(1/2)=\gamma$, $h(1)=\delta$ and

  * •

$h(t)\in W$ for every $t\in[1/2,1]$.

Write $\widetilde{h}(t)=\mathcal{C}(h(t),y)$ for $t\in[0,1]$ and let
$\widehat{h}$ be the lift of $\widetilde{h}$ across $p$ starting at $e$.
Recall from the second step of the proof that
$\widehat{h}(t)=\mathcal{C}^{\prime}(h(t),y)$ for every $t\in[0,1]$. Denote by
$p_{0}$ the homeomorphism $V^{\prime}\to V$, obtained by restricting $p$.
Since
$\widehat{h}(1/2)=\mathcal{C}^{\prime}(h(1/2),y)=\mathcal{C}^{\prime}(\gamma,y)\in
V^{\prime}$ and $\widetilde{h}(t)\in V$ for every $t\in[1/2,1]$, it follows
that $\widehat{h}|_{[1/2,1]}$ and $p_{0}^{-1}\widetilde{h}|_{[1/2,1]}$ are
both lifts of $\widetilde{h}|_{[1/2,1]}$ across $p$, starting at the same
point. Thus, by uniqueness of lifts of paths,
$\widehat{h}(t)=p_{0}^{-1}\widetilde{h}(t)$ for every $t\in[1/2,1]$. In
particular,

$\mathcal{C}^{\prime}(\delta,y)=\mathcal{C}^{\prime}(h(1),y)=\widehat{h}(1)=p_{0}^{-1}\widetilde{h}(1)\in
p_{0}^{-1}(V)=V^{\prime}.$

Thus, $\mathcal{C}^{\prime}(W\times U)\subseteq V^{\prime}$, as was to be
shown.

_4th step._ We finish the proof by verifying the uniqueness part of the
theorem.

So assume that
$\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G^{\prime})$ satisfy
$p\mathcal{C}^{\prime}=\mathcal{C}=p\mathcal{C}^{\prime\prime}$. Then
$p(\mathcal{C}^{\prime}\left(\mathcal{C}^{\prime\prime}\right)^{-1})=e$ and
hence $\mathcal{C}^{\prime}\left(\mathcal{C}^{\prime\prime}\right)^{-1}$ takes
its values in the kernel $\ker(p)$ of $p$. Since the domain $\Gamma\times X$
of $\mathcal{C}^{\prime}(\mathcal{C}^{\prime\prime})^{-1}$ is connected and
$\ker(p)$ is totally disconnected, it follows that
$\mathcal{C}^{\prime}(\mathcal{C}^{\prime\prime})^{-1}$ is constant with the
value $e$. This shows that $\mathcal{C}^{\prime}=\mathcal{C}^{\prime\prime}$.
∎

By the first part of Remark 3.39, if the acting group $\Gamma$ of a minimal
flow $\mathcal{F}$ is simply connected then the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is divisible. In the following example we show that this is
no longer the case if the assumption of simple connectedness is moved from
$\Gamma$ to $X$. (This is contrary to the well known situation of
$\Gamma=\mathbb{Z}$ and $X$ compact simply connected, in which case the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}\cong C_{z}(X,\mathbb{T}^{1})\cong
C_{z}(X,\mathbb{R})$ is divisible.)

######  Example 3.40.

Consider the usual transitive action
$\mathcal{F}\colon\text{U}(2)\curvearrowright\mathbb{S}^{3}$ (see Subsection
2.5.3). The acting group $\text{U}(2)$ of $\mathcal{F}$ is not simply
connected, having an infinite cyclic fundamental group $\pi_{1}(\text{U}(2))$,
but the phase space $\mathbb{S}^{3}$ of $\mathcal{F}$ is simply connected; we
show that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is not divisible. To
this end, let $h=\det$ be the determinant function on $\text{U}(2)$. Clearly,
$h$ is a topological morphism $\text{U}(2)\to\mathbb{T}^{1}$. Moreover, since

$h\left(\begin{array}[]{ccc}e^{i2\pi t}&0\\\
0&1\end{array}\right)=\det\left(\begin{array}[]{ccc}e^{i2\pi t}&0\\\
0&1\end{array}\right)=e^{i2\pi t}$

for every $t\in[0,1]$, the morphism
$h^{\sharp}\colon\pi_{1}(\text{U}(2))\to\pi_{1}(\mathbb{T}^{1})$ induced by
$h$ is an epimorphism (in fact, $h^{\sharp}$ is an isomorphism). Now let
$\mathcal{C}=\mathcal{Q}_{h}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ be the
quasi-coboundary induced by $h$; we claim that $\mathcal{C}$ is a non-
divisible element of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$. If $z$ is a base
point for $\mathbb{S}^{3}$ then, by virtue of (2.30) from Lemma 2.17,

$\mathcal{C}^{\sharp}\pi_{1}(\text{U}(2)\times\mathbb{S}^{3})=\mathcal{C}_{z}^{\sharp}\pi_{1}(\text{U}(2))=h^{\sharp}\pi_{1}(\text{U}(2))=\pi_{1}(\mathbb{T}^{1}).$

It follows that $\mathcal{C}$ does not lift across any of the endomorphisms
$\kappa_{d}$ of $\mathbb{T}^{1}$ with $d\geq 2$. (Indeed, recall that
$\kappa_{d}^{\sharp}\pi_{1}(\mathbb{T}^{1})=d(\pi_{1}(\mathbb{T}^{1}))\subsetneq\pi_{1}(\mathbb{T}^{1})$
for $d\geq 2$.) Thus, $\mathcal{C}$ is a non-divisible element of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, as was to be shown.

######  Corollary 3.41.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ simply connected and with $X$ compact second
countable. Let $G\in\mathsf{CAGp}$ be a connected second countable group. Then
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is the additive topological group of a
real separable Fréchet space. If the group $G$ is finite-dimensional then
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is the additive topological group of a
real separable Banach space.

######  Remark 3.42.

In the proof of the corollary we shall observe that $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})$ is a separable Banach space and find out that
it is isomorphic to $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ as a topological
group. This will yield the structure of a separable Banach space in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Now assume that the space $X$ is
infinite, so that $C_{z}(X,\mathbb{R})$ is an infinite-dimensional real linear
space. Since $\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$ contains
$\mathop{\rm{\bf B}_{\mathcal{F}}}(\mathbb{R})\cong C_{z}(X,\mathbb{R})$ as a
linear subspace, it follows from the Mackey theorem that $\mathop{\rm
dim_{\mathbb{R}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R}))\geq\mathfrak{c}$. On the other hand, by
separability of $\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$, $\mathop{\rm
dim_{\mathbb{R}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R}))\leq\mathop{\rm card}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R}))\leq\mathfrak{c}$. Consequently,

$\mathop{\rm dim_{\mathbb{R}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\mathop{\rm dim_{\mathbb{R}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R}))=\mathfrak{c}.$

###### Proof of Corollary 3.41.

Throughout the whole proof, all the groups $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ are assumed to carry the topology of u.c.s. convergence.
Let $p\colon\mathbb{R}\to\mathbb{T}^{1}$ be the usual covering morphism and
$\widehat{p}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})\ni\mathcal{D}\mapsto
p\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ be the induced topological
morphism. By virtue of Theorem 3.38, $\widehat{p}$ is an epimorphism. Since
$\Gamma\times X$ is connected and $\ker(p)\cong\mathbb{Z}$ is totally
disconnected, $\widehat{p}$ is also a monomorphism.

We show that $\widehat{p}$ is in fact a topological isomorphism. To this end,
let $(\mathcal{D}_{i})_{i\in J}$ be a net in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})$ with
$\widehat{p}(\mathcal{D}_{i})=p\mathcal{D}_{i}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}1$ in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$; we show
that $\mathcal{D}_{i}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}0$ in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$. So let
$F\subseteq\Gamma\times X$ be a compact set and $0<\varepsilon<1/2$. Fix
compact sets $1\in K\subseteq\Gamma$ and $C\subseteq X$ with $K$ connected and
with $F\subseteq K\times C$. Since $p\mathcal{D}_{i}\stackrel{{\scriptstyle
ucs}}{{\longrightarrow}}1$ in $\mathop{\rm{\bf Z}_{\mathcal{F}}}$, there is
$i_{0}\in J$ such that $p\mathcal{D}_{i}(\gamma,x)\in
p(-\varepsilon,\varepsilon)$ for all $i\geq i_{0}$, $\gamma\in K$ and $x\in
C$. Since $1\in K$ is connected and $\varepsilon<1/2$, it follows that
$\mathcal{D}_{i}(K\times x)\subseteq(-\varepsilon,\varepsilon)$ for all $i\geq
i_{0}$ and $x\in C$. Consequently,
$\mathcal{D}_{i}(F)\subseteq\mathcal{D}_{i}(K\times
C)\subseteq(-\varepsilon,\varepsilon)$ for every $i\geq i_{0}$. Hence,
$\mathcal{D}_{i}\stackrel{{\scriptstyle ucs}}{{\longrightarrow}}0$ in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$, as was to be shown.

By our assumptions on $\Gamma$ and $X$, $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})$ is the additive topological group of a real
separable Banach space, see Lemma 2.18 and Remark 2.19 in Subsection 2.5.11.
By the first part of the proof, it follows that so is the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Moreover, the groups $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ and $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ are topologically isomorphic due to Theorem 3.1 for every
$G\in\mathsf{CAGp}$. Now, if $G\in\mathsf{CAGp}$ is a connected second
countable group then its dual group $G^{*}$ is torsion-free countable and so
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)\cong\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})$ is the additive topological
group of a real separable Fréchet space by the first part of Lemma 2.1. If, in
addition, the group $G$ is finite-dimensional then the group $G^{*}$ is of a
finite rank and so $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is the additive
topological group of a real separable Banach space by the second part of Lemma
2.1. ∎

######  Theorem 3.43.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{CLAC}$ simply connected and with $X$ compact. Then
there is a short exact sequence of abelian groups

$0\longrightarrow\pi^{1}(X)\longrightarrow\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R})\longrightarrow\mathop{\rm{\bf
H}_{\mathcal{F}}}\longrightarrow 0.$ (3.17)

Further, there is a divisible torsion-free subgroup
$\mathfrak{D}\cong\mathop{\rm{\bf H}_{\mathcal{F}}}/\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$ of the groupoid $\mathop{\rm{\bf
MinH}_{\mathcal{F}}}$ with $\mathop{\rm{\bf H}_{\mathcal{F}}}=\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})\oplus\mathfrak{D}$. Finally, if
$1\leq\mathop{\rm rank}(\pi^{1}(X))<\mathfrak{c}$ then

$\mathop{\rm rank}(\mathfrak{D})=\mathop{\rm
dim_{\mathbb{Q}LS}}(\mathfrak{D})\geq\mathfrak{c}.$

######  Remark 3.44.

Notice the following facts.

  * •

The assumption $\mathop{\rm rank}(\pi^{1}(X))<\mathfrak{c}$ is satisfied if
the group $\pi^{1}(X)$ is countable. This is the case if the space $X$ is
second countable.

  * •

It follows from the last two statements of the theorem that the groupoid
$\mathop{\rm{\bf MinH}_{\mathcal{F}}}$ contains an isomorphic copy of the
group $\mathbb{Q}^{(\mathfrak{c})}\cong\mathbb{R}$.

###### Proof of Theorem 3.43.

We verify each of the three statements of the theorem in a separate step of
the proof.

_1st step._ We show that there is an exact sequence of abelian groups (3.17).

Let $p\colon\mathbb{R}\to\mathbb{T}^{1}$ be the usual covering morphism and
$\widehat{p}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})\ni\mathcal{D}\mapsto
p\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ be the induced morphism. By
an argument similar to that from Corollary 3.41, $\widehat{p}$ is an
isomorphism of groups. Further, fix a base point $z$ for $X$ and consider the
map $\varphi\colon C_{z}(X,\mathbb{T}^{1})\ni\xi\mapsto\mathop{\rm
co}(\xi)\in\mathop{\rm{\bf B}_{\mathcal{F}}}$. Clearly, $\varphi$ is an
isomorphism of groups. Finally, since the space $X$ is connected and the
kernel $\ker(p)=\mathbb{Z}$ of $p$ is totally disconnected, it follows that
$q\colon C_{z}(X,\mathbb{R})\ni\zeta\mapsto p\zeta\in C_{z}(X,\mathbb{T}^{1})$
is a monomorphism of groups. Moreover, by definition of $\pi^{1}(X)$, there is
a direct sum $C_{z}(X,\mathbb{T}^{1})=\mathop{\rm im}(q)\oplus\pi^{1}(X)$. An
elementary argument also shows that $\varphi(\mathop{\rm
im}(q))=\widehat{p}(\mathop{\rm{\bf B}_{\mathcal{F}}}(\mathbb{R}))$.

Now, the ideas formulated in the preceding paragraph yield a direct sum

$\begin{split}\widehat{p}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})&=\widehat{p}^{-1}\varphi\left(C_{z}(X,\mathbb{T}^{1})\right)=\widehat{p}^{-1}\varphi\left(\mathop{\rm
im}(q)\oplus\pi^{1}(X)\right)\\\ &=\widehat{p}^{-1}\varphi(\mathop{\rm
im}(q))\oplus\widehat{p}^{-1}\varphi(\pi^{1}(X))=\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})\oplus\widehat{p}^{-1}\varphi(\pi^{1}(X)).\end{split}$
(3.18)

Further, since $\mathop{\rm{\bf B}_{\mathcal{F}}}(\mathbb{R})$ is a linear
subspace of $\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$, it follows that
$\mathop{\rm{\bf B}_{\mathcal{F}}}(\mathbb{R})$ is a direct summand in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$ and each of its complementary
summands in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$ is isomorphic to
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})/\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})=\mathop{\rm{\bf H}_{\mathcal{F}}}(\mathbb{R})$.
Since $\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})\cap\widehat{p}^{-1}\varphi(\pi^{1}(X))=0$ by
virtue of (3.18), there is a complementary summand $H$ of $\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})$ in $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})$ with
$\widehat{p}^{-1}\varphi(\pi^{1}(X))\subseteq H$. Consequently, there are
isomorphisms of groups

$\begin{split}\mathop{\rm{\bf H}_{\mathcal{F}}}&=\mathop{\rm{\bf
Z}_{\mathcal{F}}}/\mathop{\rm{\bf
B}_{\mathcal{F}}}\cong\widehat{p}^{-1}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})/\widehat{p}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})=\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{R})/\widehat{p}^{-1}(\mathop{\rm{\bf
B}_{\mathcal{F}}})\\\ &=\left(\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})\oplus H\right)/\left(\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathbb{R})\oplus\widehat{p}^{-1}\varphi(\pi^{1}(X))\right)\cong
H/\widehat{p}^{-1}\varphi(\pi^{1}(X)).\end{split}$

Since $H\cong\mathop{\rm{\bf H}_{\mathcal{F}}}(\mathbb{R})$ and
$\widehat{p}^{-1}\varphi(\pi^{1}(X))\cong\pi^{1}(X)$, this verifies the
existence of the desired exact sequence (3.17).

_2nd step._ We verify the second statement of the theorem.

By the first step of the proof, there is an isomorphism $\mathop{\rm{\bf
Z}_{\mathcal{F}}}\cong\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$. Since
the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{R})$ is divisible, it
follows that so are the groups $\mathop{\rm{\bf Z}_{\mathcal{F}}}$,
$\mathop{\rm{\bf H}_{\mathcal{F}}}$ and $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$. Consequently, $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ is a direct summand in $\mathop{\rm{\bf H}_{\mathcal{F}}}$
and each of its complementary summands is divisible, torsion-free and
isomorphic to $\mathop{\rm{\bf H}_{\mathcal{F}}}/\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})$. Fix such a complementary summand
$\mathfrak{D}$; we need to show that $\mathfrak{D}\subseteq\mathop{\rm{\bf
MinH}_{\mathcal{F}}}$. This follows immediately from the identity
$\mathop{\rm{\bf MinH}_{\mathcal{F}}}\setminus 1=\mathop{\rm{\bf
H}_{\mathcal{F}}}\setminus\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$, see Corollary 3.9.

_3rd step._ We finish the proof of the theorem by verifying its last
statement.

So assume that $1\leq\mathop{\rm rank}(\pi^{1}(X))<\mathfrak{c}$ and fix a
complementary summand $\mathfrak{D}$ of $\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}$. By the first step
of the proof, we may view $\pi^{1}(X)$ as a subgroup of $\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R})$ in such a way that the quotient group
$\mathop{\rm{\bf H}_{\mathcal{F}}}(\mathbb{R})/\pi^{1}(X)$ is isomorphic to
$\mathop{\rm{\bf H}_{\mathcal{F}}}$. Let $A$ be the divisible hull of
$\pi^{1}(X)$ in $\mathop{\rm{\bf H}_{\mathcal{F}}}(\mathbb{R})$ and $B$ be a
complementary summand of $A$ in $\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R})$. Then $A/\pi^{1}(X)$ is a torsion group and $B$
is a torsion-free group. Since $\mathop{\rm{\bf
H}_{\mathcal{F}}}\cong\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R})/\pi^{1}(X)\cong(A/\pi^{1}(X))\oplus B$, it
follows that $A/\pi^{1}(X)\cong\mathop{\rm tor}(\mathop{\rm{\bf
H}_{\mathcal{F}}})$ and $B\cong\mathop{\rm{\bf H}_{\mathcal{F}}}/\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})\cong\mathfrak{D}$. We claim that the
following statements hold:

  * •

$\mathop{\rm rank}(\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R}))\geq\mathfrak{c}$; indeed, $\mathop{\rm{\bf
H}_{\mathcal{F}}}(\mathbb{R})$ is a real linear space, which contains a non-
trivial group $\pi^{1}(X)$,

  * •

$\mathop{\rm rank}(A)<\mathfrak{c}$; indeed, by the assumptions of the
theorem,

$\mathop{\rm rank}(A)=\mathop{\rm rank}(\pi^{1}(X))<\mathfrak{c}.$

Thus, since $\mathop{\rm{\bf H}_{\mathcal{F}}}(\mathbb{R})=A\oplus B$ and
$B\cong\mathfrak{D}$, we have $\mathop{\rm rank}(\mathfrak{D})=\mathop{\rm
rank}(B)\geq\mathfrak{c}$, as was to be shown. ∎

##  Chapter 4 Topological-Algebraic Aspects

###  4.1 Free objects for categories of maps

Our aim in this section is to prove the existence of free objects for certain
categories of continuous base point preserving maps from a given pointed space
into compact abelian groups. Our results from this section will be applied in
the next section, where we show the existence of a free compact abelian group
extension for an arbitrary minimal flow.

Let $X$ be a pointed space with the base point $z$ and $A$ be a subgroup of
the group $C_{z}(X,\mathbb{T}^{1})$ of all continuous base point preserving
maps $X\to\mathbb{T}^{1}$. Given $G\in\mathsf{CAGp}$, let $A_{G}$ denote the
set of all $f\in C_{z}(X,G)$ such that $\chi f\in A$ for every $\chi\in
G^{*}$. Clearly, $A_{G}$ is a subgroup of $C_{z}(X,G)$. Moreover, if
$G,H\in\mathsf{CAGp}$ and $h\in\mathop{\rm Hom}(G,H)$, then $hf\in A_{H}$ for
every $f\in A_{G}$. Now consider the category $\mathsf{Maps}_{A}$ defined as
follows. The objects of $\mathsf{Maps}_{A}$ are the elements of $A_{G}$ with
$G\in\mathsf{CAGp}$ arbitrary. For $f_{1}\in A_{G_{1}}$ and $f_{2}\in
A_{G_{2}}$, the set of morphisms $\mathop{\rm Hom}(f_{1},f_{2})$ in
$\mathsf{Maps}_{A}$ consists of all $h\in\mathop{\rm Hom}(G_{1},G_{2})$ with
$hf_{1}=f_{2}$. The composition of morphisms is obtained by restricting that
from $\mathsf{CAGp}$. Clearly, $\mathop{\rm id}_{G}\in\mathop{\rm Hom}(f,f)$
for all $f\in A_{G}$ and $G\in\mathsf{CAGp}$.

Our goal now is to show that $\mathsf{Maps}_{A}$ is a category with a free
object. This is done in Theorem 4.1 below and some examples are given in
Example 4.2. Recall that a free object for $\mathsf{Maps}_{A}$ consists of a
group $G_{A}\in\mathsf{CAGp}$ and of a map $f_{A}\in A_{G_{A}}$ with the
following universal property:

  * •

for every $G\in\mathsf{CAGp}$ and every $f\in A_{G}$ there is a unique
$h\in\mathop{\rm Hom}(G_{A},G)$ with $hf_{A}=f$.

######  Theorem 4.1.

Let $X$ be a pointed space and $A$ be a subgroup of $C_{z}(X,\mathbb{T}^{1})$
equipped with the discrete topology. Set $G_{A}=A^{*}$ and let $f_{A}\colon
X\to G_{A}$ be given by the rule $f_{A}(x)\colon
A\ni\varphi\mapsto\varphi(x)\in\mathbb{T}^{1}$ for every $x\in X$. Then
$G_{A}$ and $f_{A}$ constitute a free object for the category
$\mathsf{Maps}_{A}$.

###### Proof.

The group $G_{A}$ is compact abelian, being a dual group of a discrete abelian
group $A$. The map $f_{A}$ is well defined (that is, $f_{A}(x)\in A^{*}$ for
every $x\in X$), it is base point preserving and continuous. It therefore
remains to verify the universality of $f_{A}$. That is, we must show that for
every $G\in\mathsf{CAGp}$ and every $f\in A_{G}$ there is a unique
$h\in\mathop{\rm Hom}(G_{A},G)$ with $hf_{A}=f$. Since the latter condition is
an invariant of topological isomorphism with respect to $G$, we may assume
that $G=D^{*}$ for an appropriate group $D\in\mathsf{DAGp}$. Thus, we need to
show that $k^{*}f_{A}=f$ for one and only one $k\in\mathop{\rm Hom}(D,A)$. The
uniqueness of $k$ is clear, for the equality $k^{*}f_{A}=f$ translates into
$k(d)(x)=f(x)(d)$ for all $d\in D$ and $x\in X$, and this in turn is satisfied
for at most one function $k\colon D\to A$. We finish the proof by showing that
$k$ thus defined is a (well defined) morphism of groups.

To see that $k$ takes its values in $A$, fix $d\in D$ and consider the
character $\chi$ of $D^{*}$ given by $\chi(\Upsilon)=\Upsilon(d)$ for every
$\Upsilon\in D^{*}$. Then $\chi f=k(d)$. Since $f\in A_{D^{*}}$, it follows
that $\chi f\in A$ and hence $k(d)\in A$. The fact that $k$ is a morphism of
groups $D\to A$ follows at once from the fact that $f(x)$ is a morphism of
groups $D\to\mathbb{T}^{1}$ for every $x\in X$. ∎

######  Example 4.2.

We present three examples of categories $\mathsf{Maps}_{A}$ and the ranges
$G_{A}$ of their free objects $f_{A}$. We shall abuse the terminology here and
refer to the groups $G_{A}$ as the free objects for the categories
$\mathsf{Maps}_{A}$.

  * ($\iota$)

Let $X$ be a pointed space and $A$ be the whole group
$C_{z}(X,\mathbb{T}^{1})$. Then $\mathsf{Maps}_{A}$ is the category of all
continuous base point preserving maps $X\to G$ with $G\in\mathsf{CAGp}$ and
the free object $G_{A}$ for $\mathsf{Maps}_{A}$ is the free compact abelian
group over $X$. If, for instance, the space $X$ is compact connected second
countable and non-degenerate then $C_{z}(X,\mathbb{R})$ is a real linear space
with dimension $\mathop{\rm
dim_{\mathbb{R}LS}}(C_{z}(X,\mathbb{R}))=\mathop{\rm
w}(X)^{\aleph_{0}}=(\aleph_{0})^{\aleph_{0}}=\mathfrak{c}$, and hence there
are topological isomorphisms

$\begin{split}G_{A}&=A^{*}=\left(C_{z}(X,\mathbb{T}^{1})_{d}\right)^{*}\cong\left(C_{z}(X,\mathbb{R})_{d}\oplus\pi^{1}(X)\right)^{*}\\\
&\cong\left(\mathbb{R}_{d}^{(\mathfrak{c})}\oplus\pi^{1}(X)\right)^{*}\cong\left(\mathbb{R}_{d}\oplus\pi^{1}(X)\right)^{*}\\\
&\cong b\mathbb{R}\times\pi^{1}(X)^{*}.\end{split}$

  * ($\iota\iota$)

Let $X$ be a non-degenerate compact connected second countable pointed space
and $A$ be the subgroup of $C_{z}(X,\mathbb{T}^{1})$ formed by the null-
homotopic maps. Then $A$ is the identity arc-component of
$C_{z}(X,\mathbb{T}^{1})$ and it is naturally isomorphic to
$C_{z}(X,\mathbb{R})$. The category $\mathsf{Maps}_{A}$ consists of the
continuous base point preserving null-homotopic maps $X\to G$ with
$G\in\mathsf{CAGp}$. Similarly to ($\iota$), there are topological
isomorphisms

$G_{A}=A^{*}\cong\left(C_{z}(X,\mathbb{R})_{d}\right)^{*}\cong\left(\mathbb{R}_{d}^{(\mathfrak{c})}\right)^{*}\cong(\mathbb{R}_{d})^{*}\cong
b\mathbb{R}.$

  * ($\iota\iota\iota$)

Let $X$ be a compact abelian group and $A=X^{*}$ be the dual group of $X$.
Clearly, $A$ is a subgroup of $C_{z}(X,\mathbb{T}^{1})$. Since the characters
of compact abelian groups separate points, we have $A_{G}=\mathop{\rm
Hom}(X,G)$ for every $G\in\mathsf{CAGp}$ and $\mathsf{Maps}_{A}$ is thus the
category of the (topological) morphisms $X\to G$ with $G\in\mathsf{CAGp}$. The
free object $G_{A}$ of $\mathsf{Maps}_{A}$ is given by the topological
isomorphism $G_{A}=A^{*}=X^{**}\cong X$.

######  Corollary 4.3.

Under the assumptions of Theorem 4.1, the category $\mathsf{Maps}_{A}$ is
isomorphic to the category $\mathsf{Hom}(G_{A},\mathsf{CAGp})$.

###### Proof.

We shall define a covariant functor
$\mathsf{F}\colon\mathsf{Maps}_{A}\to\mathsf{Hom}(G_{A},\mathsf{CAGp})$ and
show that it is an isomorphism of categories. Given $G\in\mathsf{CAGp}$ and
$f\in A_{G}$, we let $\mathsf{F}(f)$ be the unique element of $\mathop{\rm
Hom}(G_{A},G)$ with $\mathsf{F}(f)f_{A}=f$. By universality of $f_{A}$, this
defines a bijection between $A_{G}$ and $\mathop{\rm Hom}(G_{A},G)$. Further,
given $G_{1},G_{2}\in\mathsf{CAGp}$, $f_{1}\in A_{G_{1}}$, $f_{2}\in
A_{G_{2}}$ and $l\in\mathop{\rm Hom}(f_{1},f_{2})$, we set
$\mathsf{F}(l)=l\in\mathop{\rm Hom}(\mathsf{F}(f_{1}),\mathsf{F}(f_{2}))$.
This definition is correct, for $l\in\mathop{\rm Hom}(G_{1},G_{2})$ and
$lF(f_{1})=F(f_{2})$; indeed, $(lF(f_{1}))f_{A}=lf_{1}=f_{2}=F(f_{2})f_{A}$
and hence, by universality of $f_{A}$, $lF(f_{1})=F(f_{2})$. Moreover, we see
at once that $\mathsf{F}$ defines a bijection between $\mathop{\rm
Hom}(f_{1},f_{2})$ and $\mathop{\rm
Hom}(\mathsf{F}(f_{1}),\mathsf{F}(f_{2}))$. The functorial properties
$\mathsf{F}(l_{2}l_{1})=\mathsf{F}(l_{2})\mathsf{F}(l_{1})$ and
$\mathsf{F}(\mathop{\rm id}_{f})=\mathop{\rm id}_{\mathsf{F}(f)}$ are
straightforward and so $\mathsf{F}$ is indeed an isomorphism of categories. ∎

###  4.2 Free group extensions

Our aim in this section is to show that for every minimal flow $\mathcal{F}$,
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is a category with a free object.
This free object consists of a group
$\mathop{\rm\gimel_{\mathcal{F}}}\in\mathsf{CAGp}$ and of an extension
$\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}})$, and is universal in the
sense that every object from $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is
obtained as a composition of $\mathop{\rm\daleth_{\mathcal{F}}}$ with a unique
topological morphism. The existence of a free object in
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is interesting from the categorical
point of view, for it shows that $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$
can be viewed as a category of morphisms of compact abelian groups. However,
from the point of view of topological dynamics this fact becomes even more
important: the dynamics of every extension
$\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is encoded in the
dynamics of $\mathop{\rm\daleth_{\mathcal{F}}}$.

After showing that the free extension
$(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ of
$\mathcal{F}$ exists, we compute $\mathop{\rm\gimel_{\mathcal{F}}}$ in the
case when $\Gamma$ is a simply connected Lie group and describe relationships
of $(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ to
some of the concepts introduced earlier. The existence of a free object for
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ becomes even more valuable to us in
Section 4.4, where we show that a free object for the category of the minimal
extensions $\mathcal{C}\in\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ typically
does not exist.

######  Definition 4.4.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
The _free compact abelian group extension_ (briefly, the _free extension_) of
$\mathcal{F}$ is defined as the free object for the category
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. Recall that it consists of a group
$\mathop{\rm\gimel_{\mathcal{F}}}\in\mathsf{CAGp}$ and an extension
$\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}})$ with the following
universal property:

  * •

for every $G\in\mathsf{CAGp}$ and every $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ there exists a unique $h_{\mathcal{C}}\in\mathop{\rm
Hom}(\mathop{\rm\gimel_{\mathcal{F}}},G)$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$.

The free extension of $\mathcal{F}$ is of course unique up to a natural
isomorphism. On occasions we shall take the liberty of abusing the terminology
by referring either to $\mathop{\rm\gimel_{\mathcal{F}}}$ or to
$\mathop{\rm\daleth_{\mathcal{F}}}$ as the free extension of $\mathcal{F}$.

######  Theorem 4.5.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Equip the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with the discrete topology
and define $\mathop{\rm\gimel_{\mathcal{F}}}$ as its Pontryagin dual

$\mathop{\rm\gimel_{\mathcal{F}}}=\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}\right)_{d}^{*}.$

Further, let $\mathop{\rm\daleth_{\mathcal{F}}}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}})$ be given by

$\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}\ni\mathcal{C}\mapsto\mathcal{C}(\gamma,x)\in\mathbb{T}^{1}$

for all $\gamma\in\Gamma$ and $x\in X$. Then
$(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ is the
free extension of $\mathcal{F}$.

###### Proof.

Fix $z\in X$ and set $A=\mathop{\rm{\bf Z}_{\mathcal{F}}}$. Then $A$ is a
subgroup of $C_{(1,z)}(\Gamma\times X,\mathbb{T}^{1})$. Moreover, under the
notation from Section 4.1, we have $A_{G}=\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ for every $G\in\mathsf{CAGp}$, for the characters of
compact abelian groups separate points. Consequently,
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}=\mathsf{Maps}_{A}$ and the statement
of the theorem thus follows directly from Theorem 4.1. ∎

######  Example 4.6.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
with $\Gamma\in\mathsf{LieGp}$ simply connected and with $X$ compact second
countable. Assume that both $\Gamma$ and $X$ are infinite. We show that there
is a topological isomorphism $\mathop{\rm\gimel_{\mathcal{F}}}\cong
b\mathbb{R}$. Indeed, by virtue of Corollary 3.41 and Remark 3.42,
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is a real linear space with dimension
$\mathop{\rm dim_{\mathbb{R}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\mathfrak{c}$. Consequently,

$\mathop{\rm\gimel_{\mathcal{F}}}=\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}\right)_{d}^{*}\cong\left(\mathbb{R}^{(\mathfrak{c})}\right)_{d}^{*}\cong\left(\mathbb{R}_{d}\right)^{*}=b\mathbb{R}.$

######  Theorem 4.7.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ be
the model of the free extension of $\mathcal{F}$ constructed in Theorem 4.5.
Then the following statements hold.

  1. (1)

The category $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ is isomorphic to the
category $\mathsf{Hom}(\mathop{\rm\gimel_{\mathcal{F}}},\mathsf{CAGp})$.

  2. (2)

Under the identification
$(\mathop{\rm\gimel_{\mathcal{F}}})^{*}=(\mathop{\rm{\bf
Z}_{\mathcal{F}}})_{d}^{**}\cong\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}\right)_{d}$ it holds

$F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=\mathop{\rm{\bf
B}_{\mathcal{F}}}\hskip 11.38109pt\text{and}\hskip
11.38109pt\left(\mathop{\rm\daleth_{\mathcal{F}}}\right)^{*}=\emph{Id}_{\mathop{\rm{\bf
Z}_{\mathcal{F}}}}.$

  3. (3)

Let
$\mathop{\rm\gimel_{\mathcal{F}}^{c}}=\mathop{\rm\gimel_{\mathcal{F}}}/F(\mathop{\rm\daleth_{\mathcal{F}}})$,
$\kappa\colon\mathop{\rm\gimel_{\mathcal{F}}}\to\mathop{\rm\gimel_{\mathcal{F}}^{c}}$
be the quotient morphism,
$\mathop{\rm\daleth_{\mathcal{F}}^{c}}=\kappa\mathop{\rm\daleth_{\mathcal{F}}}$,
$z$ be the base point of $X$ and $\vartheta$ be the base point preserving
transfer function for $\mathop{\rm\daleth_{\mathcal{F}}^{c}}$. Then
$\mathop{\rm\gimel_{\mathcal{F}}^{c}}$ is the free compact abelian group over
$X$ with respect to the universal map $\vartheta$.

  4. (4)

For every $G\in\mathsf{CAGp}$ there is an isomorphism of groups

$\widetilde{\Phi}_{G}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\ni\mathcal{C}\mapsto\left(\omega_{G}\right)^{-1}\Phi_{G}(\mathcal{C})^{*}=h_{\mathcal{C}}\in\mathop{\rm
Hom}\left(\mathop{\rm\gimel_{\mathcal{F}}},G\right),$

where $\omega_{G}$ is the Pontryagin isomorphism $G\to G^{**}$,
$\Phi_{G}(\mathcal{C})^{*}=\mathcal{C}^{**}$ stands for the morphism dual to
$\Phi_{G}(\mathcal{C})=\mathcal{C}^{*}$ and the morphism $h_{\mathcal{C}}$ is
defined by the formula
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$.

######  Remark 4.8.

We wish to make the following observations.

  * •

Under the identification from statement (2), $\mathop{\rm{\bf
B}_{\mathcal{F}}}\subseteq\mathop{\rm{\bf Z}_{\mathcal{F}}}$ corresponds to
$(\mathop{\rm{\bf B}_{\mathcal{F}}})^{\perp\perp}\subseteq(\mathop{\rm{\bf
Z}_{\mathcal{F}}})_{d}^{**}=(\mathop{\rm\gimel_{\mathcal{F}}})^{*}$.
Consequently, the isomorphism
$F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}\cong\mathop{\rm{\bf
B}_{\mathcal{F}}}$ yields an equality
$F(\mathop{\rm\daleth_{\mathcal{F}}})=(\mathop{\rm{\bf
B}_{\mathcal{F}}})^{\perp}$ and a topological isomorphism
$F(\mathop{\rm\daleth_{\mathcal{F}}})\cong(\mathop{\rm{\bf
Z}_{\mathcal{F}}}/\mathop{\rm{\bf B}_{\mathcal{F}}})_{d}^{*}=(\mathop{\rm{\bf
H}_{\mathcal{F}}})_{d}^{*}$. Under this isomorphism the identity component
$F(\mathop{\rm\daleth_{\mathcal{F}}})_{0}$ of
$F(\mathop{\rm\daleth_{\mathcal{F}}})$ corresponds to the group $\mathop{\rm
tor}(\mathop{\rm{\bf H}_{\mathcal{F}}})^{\perp}$. Consequently, the group
$F(\mathop{\rm\daleth_{\mathcal{F}}})$ is connected if and only if the group
$\mathop{\rm{\bf H}_{\mathcal{F}}}$ is torsion-free. Similarly,
$F(\mathop{\rm\daleth_{\mathcal{F}}})$ is torsion-free if and only if
$\mathop{\rm{\bf H}_{\mathcal{F}}}$ is divisible.

  * •

Recall that
$\mathop{\rm\gimel_{\mathcal{F}}}/F(\mathop{\rm\daleth_{\mathcal{F}}})=\mathop{\rm\gimel_{\mathcal{F}}^{c}}$
is topologically isomorphic to the right-hand side of (2.17) if the base space
$X$ of the flow $\mathcal{F}$ is compact. If $X$ is additionally assumed
connected (which is the case if $\Gamma$ is connected) then the middle term of
the right-hand side of (2.17) vanishes.

###### Proof of Theorem 4.7.

Statement (1) follows from Theorem 4.1, Corollary 4.3 and Theorem 4.5, for
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}=\mathsf{Maps}_{A}$ with
$A=\mathop{\rm{\bf Z}_{\mathcal{F}}}$.

In order to verify statement (2), we adopt the following notation. Given
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$, let
$\Upsilon_{\mathcal{C}}$ be the image of $\mathcal{C}$ under the Pontryagin
isomorphism

$\omega_{(\mathop{\rm{\bf Z}_{\mathcal{F}}})_{d}}\colon(\mathop{\rm{\bf
Z}_{\mathcal{F}}})_{d}\to(\mathop{\rm{\bf
Z}_{\mathcal{F}}})_{d}^{**}=(\mathop{\rm\gimel_{\mathcal{F}}})^{*}.$

Then $\Upsilon_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$.
From this observation and from the properties of the functor $F$ we obtain the
equivalence of the following conditions for every
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$:

  * •

$\Upsilon_{\mathcal{C}}\in F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}$,

  * •

$\Upsilon_{\mathcal{C}}F(\mathop{\rm\daleth_{\mathcal{F}}})=1$,

  * •

$F(\Upsilon_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}})=1$,

  * •

$F(\mathcal{C})=1$,

  * •

$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}$.

This verifies the first statement from (2). As for the second statement from
(2), for every $\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$,
$(\mathop{\rm\daleth_{\mathcal{F}}})^{*}(\Upsilon_{\mathcal{C}})=\Upsilon_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$,
and so the action of $(\mathop{\rm\daleth_{\mathcal{F}}})^{*}$ on
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is indeed identical.

We verify statement (3). First observe that
$\mathop{\rm\daleth_{\mathcal{F}}^{c}}=\kappa\mathop{\rm\daleth_{\mathcal{F}}}$
is a coboundary and the map $\vartheta$ is thus well defined; indeed, we have
$F(\mathop{\rm\daleth_{\mathcal{F}}^{c}})=F(\kappa\mathop{\rm\daleth_{\mathcal{F}}})=\kappa
F(\mathop{\rm\daleth_{\mathcal{F}}})=e$ and hence
$\mathop{\rm\daleth_{\mathcal{F}}^{c}}\in\mathop{\rm{\bf
B}_{\mathcal{F}}}(\mathop{\rm\gimel_{\mathcal{F}}^{c}})$. Now let
$G\in\mathsf{CAGp}$ and $\xi\colon X\to G$ be a continuous base point
preserving map; we need to show that there is a unique topological morphism
$\sigma\colon\mathop{\rm\gimel_{\mathcal{F}}^{c}}\to G$ with
$\sigma\vartheta=\xi$. By universality of $\mathop{\rm\daleth_{\mathcal{F}}}$,
there exists a unique $h\in\mathop{\rm
Hom}(\mathop{\rm\gimel_{\mathcal{F}}},G)$ with
$h\mathop{\rm\daleth_{\mathcal{F}}}=\mathop{\rm co}(\xi)$. Since
$e=F(\mathop{\rm
co}(\xi))=F(h\mathop{\rm\daleth_{\mathcal{F}}})=hF(\mathop{\rm\daleth_{\mathcal{F}}})$,
$h$ factors through $\kappa$ onto a morphism
$\sigma\colon\mathop{\rm\gimel_{\mathcal{F}}^{c}}\to G$; that is,
$\sigma\kappa=h$. Thus,

$\mathop{\rm
co}(\xi)=h\mathop{\rm\daleth_{\mathcal{F}}}=\sigma\kappa\mathop{\rm\daleth_{\mathcal{F}}}=\sigma\mathop{\rm\daleth_{\mathcal{F}}^{c}}=\sigma\mathop{\rm
co}(\vartheta)=\mathop{\rm co}(\sigma\vartheta)$

and, since both $\xi$ and $\sigma\vartheta$ are base point preserving maps, it
follows from minimality of $\mathcal{F}$ that $\xi=\sigma\vartheta$. To verify
the uniqueness of $\sigma$ with the latter property assume that
$\varrho\in\mathop{\rm Hom}(\mathop{\rm\gimel_{\mathcal{F}}^{c}},G)$ also
satisfies $\varrho\vartheta=\xi$. Then

$\sigma\kappa\mathop{\rm\daleth_{\mathcal{F}}}=h\mathop{\rm\daleth_{\mathcal{F}}}=\mathop{\rm
co}(\xi)=\mathop{\rm co}(\varrho\vartheta)=\varrho\mathop{\rm
co}(\vartheta)=\varrho\mathop{\rm\daleth_{\mathcal{F}}^{c}}=\varrho\kappa\mathop{\rm\daleth_{\mathcal{F}}}$

and hence, by universality of $\mathop{\rm\daleth_{\mathcal{F}}}$,
$\sigma\kappa=\varrho\kappa$. Finally, since $\kappa$ is an epimorphism, it
follows that $\sigma=\varrho$. This finishes the proof of statement (3).

We verify statement (4). Fix $G\in\mathsf{CAGp}$. Recall that there is an
isomorphism of groups

$\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})\ni
k\mapsto\left(\omega_{G}\right)^{-1}k^{*}\in\mathop{\rm
Hom}(\mathop{\rm\gimel_{\mathcal{F}}},G).$

Since $\Phi_{G}$ is an isomorphism $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ by virtue of Theorem 3.1, it follows that
$\widetilde{\Phi}_{G}$ is an isomorphism $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\to\mathop{\rm Hom}(\mathop{\rm\gimel_{\mathcal{F}}},G)$.
Thus, to finish the proof of (4), it remains to show that
$(\omega_{G})^{-1}\mathcal{C}^{**}=h_{\mathcal{C}}$ or, equivalently, that
$\mathcal{C}^{**}=\omega_{G}h_{\mathcal{C}}$ for every
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. So fix
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. By universality of
$\mathop{\rm\daleth_{\mathcal{F}}}$ and by virtue of the identity
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$, the desired
equality $\mathcal{C}^{**}=\omega_{G}h_{\mathcal{C}}$ is equivalent with
$\mathcal{C}^{**}\mathop{\rm\daleth_{\mathcal{F}}}=\omega_{G}\mathcal{C}$. It
is the latter equality that we now verify. Observe that both
$\mathcal{C}^{**}\mathop{\rm\daleth_{\mathcal{F}}}$ and
$\omega_{G}\mathcal{C}$ are elements of $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G^{**})$.

Fix $\gamma\in\Gamma$, $x\in X$ and $\chi\in G^{*}$. Then

$\left(\mathcal{C}^{**}\mathop{\rm\daleth_{\mathcal{F}}}\right)(\gamma,x)\colon\chi\mapsto\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)(\mathcal{C}^{*}(\chi))=\mathop{\rm\daleth_{\mathcal{F}}}(\gamma,x)(\chi\mathcal{C})=\chi\mathcal{C}(\gamma,x)$

by definition of $\mathcal{C}^{*}$ and $\mathop{\rm\daleth_{\mathcal{F}}}$.
Likewise,

$\left(\omega_{G}\mathcal{C}\right)(\gamma,x)\colon\chi\mapsto\omega_{G}(\mathcal{C}(\gamma,x))(\chi)=\chi(\mathcal{C}(\gamma,x))=\chi\mathcal{C}(\gamma,x)$

by definition of $\omega_{G}$. This verifies the equality
$\mathcal{C}^{**}\mathop{\rm\daleth_{\mathcal{F}}}=\omega_{G}\mathcal{C}$. ∎

######  Proposition 4.9.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and $(\mathop{\rm\gimel_{\mathcal{F}}},\mathop{\rm\daleth_{\mathcal{F}}})$ be
the model of the free extension of $\mathcal{F}$ constructed in Theorem 4.5.
Then the following statements hold.

  1. ($\iota$)

For $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$ let $h_{\mathcal{C}}$ be the unique morphism
$\mathop{\rm\gimel_{\mathcal{F}}}\to G$ with
$h_{\mathcal{C}}\mathop{\rm\daleth_{\mathcal{F}}}=\mathcal{C}$ and let
$H_{\mathcal{C}}=\ker(h_{\mathcal{C}})$, $G_{\mathcal{C}}=\mathop{\rm
im}(h_{\mathcal{C}})$. Then

    1. (a)

$F(\mathcal{C})=h_{\mathcal{C}}F(\mathop{\rm\daleth_{\mathcal{F}}})$,

    2. (b)

$(h_{\mathcal{C}})^{*}=\mathcal{C}^{*}$ (under the identification
$(\mathop{\rm\gimel_{\mathcal{F}}})^{*}\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}$),

    3. (c)

$H_{\mathcal{C}}^{\perp}=\mathop{\rm im}((h_{\mathcal{C}})^{*})=\mathop{\rm
im}(\mathcal{C}^{*})$,

    4. (d)

$G_{\mathcal{C}}$ is the closed subgroup of $G$ generated by the values of
$\mathcal{C}$.

  2. ($\iota\iota$)

For all $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$, the following conditions are equivalent:

    1. (i)

$\mathcal{C}\in\mathop{\rm{\bf B}_{\mathcal{F}}}(G)$,

    2. (ii)

$F(\mathop{\rm\daleth_{\mathcal{F}}})\subseteq H_{\mathcal{C}}$.

  3. ($\iota\iota\iota$)

For all $G\in\mathsf{CAGp}$ and $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$, the following conditions are equivalent:

    1. ($\alpha$)

$\mathcal{C}$ is minimal,

    2. ($\beta$)

$G_{\mathcal{C}}=G$ and $H_{\mathcal{C}}^{\perp}\cap
F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=0$,

    3. ($\gamma$)

$G_{\mathcal{C}}=G$ and
$H_{\mathcal{C}}+F(\mathop{\rm\daleth_{\mathcal{F}}})=\mathop{\rm\gimel_{\mathcal{F}}}$.

###### Proof.

We begin by verifying ($\iota$). Statement (a) follows immediately from the
definition of $h_{\mathcal{C}}$ and from the properties of the functor $F$. To
verify statement (b), observe that
$(\mathop{\rm\daleth_{\mathcal{F}}})^{*}(h_{\mathcal{C}})^{*}=\mathcal{C}^{*}$
by definition of $h_{\mathcal{C}}$. Moreover, under the identification
$(\mathop{\rm\gimel_{\mathcal{F}}})^{*}\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}$, $(\mathop{\rm\daleth_{\mathcal{F}}})^{*}$ equals the
identity on $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ by statement (2) of Theorem
4.7. Thus, it follows that $(h_{\mathcal{C}})^{*}=\mathcal{C}^{*}$. The first
equality in (c) follows by a simple computation
$H_{\mathcal{C}}^{\perp}=\ker(h_{\mathcal{C}})^{\perp}=\mathop{\rm
im}((h_{\mathcal{C}})^{*})$ and the second equality follows from (b). Let
$V_{\mathcal{C}}$ be the closed subgroup of $G$ generated by the values of
$\mathcal{C}$. Clearly, $\chi\in G^{*}$ belongs to $V_{\mathcal{C}}^{\perp}$
if and only if $\mathcal{C}^{*}(\chi)=\chi\mathcal{C}=1$. Thus, by virtue of
(b),
$V_{\mathcal{C}}^{\perp}=\ker(\mathcal{C}^{*})=\ker((h_{\mathcal{C}})^{*})=\mathop{\rm
im}(h_{\mathcal{C}})^{\perp}=G_{\mathcal{C}}^{\perp}$. Since $V_{\mathcal{C}}$
and $G_{\mathcal{C}}$ are closed subgroups of $G$ by definition, it follows
that $G_{\mathcal{C}}=V_{\mathcal{C}}$. This verifies statement (d).

Statement ($\iota\iota$) follows at once from part (a) of statement ($\iota$)
and from Theorem 3.7(2).

We finish the proof of the proposition by verifying statement
($\iota\iota\iota$). By part (a) of statement ($\iota$), we have
$F(\mathcal{C})^{\perp}=(h_{\mathcal{C}}F(\mathop{\rm\daleth_{\mathcal{F}}}))^{\perp}=((h_{\mathcal{C}})^{*})^{-1}(F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp})$.
Thus, by part (c) of statement ($\iota$), the following conditions are
equivalent:

  * •

$\mathcal{C}$ is minimal,

  * •

$F(\mathcal{C})=G$,

  * •

$((h_{\mathcal{C}})^{*})^{-1}(F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp})=0$,

  * •

$(h_{\mathcal{C}})^{*}$ is a monomorphism and $\mathop{\rm
im}((h_{\mathcal{C}})^{*})\cap
F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=0$,

  * •

$h_{\mathcal{C}}$ is an epimorphism and $H_{\mathcal{C}}^{\perp}\cap
F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=0$,

  * •

$G_{\mathcal{C}}=G$ and $H_{\mathcal{C}}^{\perp}\cap
F(\mathop{\rm\daleth_{\mathcal{F}}})^{\perp}=0$.

This verifies the equivalence of ($\alpha$) and ($\beta$). The equivalence of
($\beta$) and ($\gamma$) follows from basic properties of the annihilator
mechanism. ∎

###  4.3 Divisibility and torsion-freeness

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
In this section we study two important properties of the groups
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ ($G\in\mathsf{CAGp}$), namely their
torsion-freeness and divisibility. First, we relate the torsion-freeness of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ to the topological(-algebraic)
properties of $X$ and $\Gamma$. Then we relate the divisibility of the groups
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ to the divisibility of the single group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and show how it is reflected in the
existence of lifts of extensions across various types of epimorphisms within
$\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$. We also gain from these results
some useful information about the free extension
$\mathop{\rm\gimel_{\mathcal{F}}}$ of $\mathcal{F}$.

######  Theorem 4.10.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then the following statements are equivalent:

  1. (i)

$\mathop{\rm\gimel_{\mathcal{F}}}$ is connected,

  2. (ii)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is torsion-free,

  3. (iii)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is torsion-free for every
$G\in\mathsf{CAGp}$,

  4. (iv)

the space $X$ is connected and the group $\Gamma$ has no non-trivial finite
abelian quotient groups.

######  Remark 4.11.

Let us mention the following facts.

  * •

In connection with (iv) observe that a group $\Gamma$ may posses a non-trivial
finite quotient group and still not possess a non-trivial finite _abelian_
quotient group. For instance, a finite non-abelian simple group (say, the
alternating group $A_{5}$) has no proper normal subgroups and hence it
possesses no abelian quotient groups other than the trivial one. Of course,
our interest is in infinite topological transformation groups, but our remark
now applies for instance to $\Gamma=\mathbb{R}\times A_{5}$.

  * •

A discrete abelian group $\Gamma$ possesses no non-trivial finite abelian
quotient groups if and only if it is divisible. This occurs if and only if its
dual group $\Gamma^{*}$ is torsion-free.

  * •

The two conditions from (iv) are automatically satisfied if the group $\Gamma$
is connected, which is the situation of our main interest.

###### Proof of Theorem 4.10.

The equivalence of (i) and (ii) is an immediate consequence of the definition
of $\mathop{\rm\gimel_{\mathcal{F}}}$. Implication (iii)$\Rightarrow$(ii) is
clear. The converse implication follows from Theorem 3.1, which yields

$\begin{split}\mathop{\rm tor}(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G))&\cong\mathop{\rm tor}(\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}}))=\bigcup_{k\in\mathbb{N}}\text{tor}_{k}(\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}}))\\\
&=\bigcup_{k\in\mathbb{N}}\mathop{\rm
Hom}(G^{*},\text{tor}_{k}(\mathop{\rm{\bf Z}_{\mathcal{F}}}))=0\end{split}$

for every $G\in\mathsf{CAGp}$.

To show that (iv)$\Rightarrow$(ii), assume that $X$ is connected and $\Gamma$
has no non-trivial finite abelian quotient groups. Fix
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}$ and $k\in\mathbb{N}$ with
$\mathcal{C}^{k}=1$. Then $\mathcal{C}$ takes its values in $\mathbb{Z}_{k}$.
By connectedness of $X$, the mapping
$l\colon\Gamma\ni\gamma\mapsto\mathcal{C}(\gamma,x)\in\mathbb{Z}_{k}$ does not
depend on the choice of $x\in X$ and from the cocycle identity for
$\mathcal{C}$ it follows that $l$ is a morphism of topological groups. It is
clearly a quotient morphism onto its image. Since the group $\Gamma$ has no
non-trivial finite abelian quotient groups by our assumptions, it follows that
$l=1$ and hence $\mathcal{C}=1$. Thus, indeed, $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is a torsion-free group.

We finish the proof by showing that (ii)$\Rightarrow$(iv). Assume that
condition (iv) is not satisfied and consider first the case of $X$ not
connected. Fix a continuous map $\xi\colon X\to\mathbb{T}^{1}$ with
$\xi(X)=\\{\pm 1\\}$. Then, by minimality of $\mathcal{F}$, $\mathop{\rm
co}(\xi)$ is a non-trivial element of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ with
$\mathop{\rm co}(\xi)^{2}=1$ and hence $\mathop{\rm tor}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})\neq\emptyset$. Now consider the case when $\Gamma$ has a
non-trivial finite abelian quotient group. Then $\Gamma$ factors onto
$\mathbb{Z}_{k}$ for some $k\geq 2$ and the quasi-coboundary induced by the
underlying epimorphism $\Gamma\to\mathbb{Z}_{k}$ is a non-trivial element of
$\mathop{\rm tor}_{k}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$. This verifies the
desired implication. ∎

######  Theorem 4.12.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then the following conditions are equivalent:

  1. (1)

$\mathop{\rm\gimel_{\mathcal{F}}}$ is torsion-free,

  2. (2)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible,

  3. (3)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is divisible for every
$G\in\mathsf{CAGp}$ connected,

  4. (4)

the objects of $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ lift across the
finite-to-one epimorphisms from $\mathsf{CAGp}$,

  5. (5)

the objects of $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ lift across all
epimorphisms from $\mathsf{CAGp}$.

  6. (6)

for every every $G\in\mathsf{CAGp}$ and every $H\mathrel{\lhd}G$, the
canonical sequence

$0\longrightarrow H\stackrel{{\scriptstyle
j}}{{\longrightarrow}}G\stackrel{{\scriptstyle
p}}{{\longrightarrow}}G/H\longrightarrow 0$

gives rise to the short exact sequence of abelian groups

$0\longrightarrow\mathop{\rm{\bf
Z}_{\mathcal{F}}}(H)\stackrel{{\scriptstyle\widehat{j}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\widehat{p}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G/H)\longrightarrow 0.$

######  Remark 4.13.

The assumption of connectedness of $G$ in condition (3) is in general not
redundant, for otherwise the morphisms $\kappa_{k}\colon G\to G$
($k\in\mathbb{N}$) would not be surjective. However, if we restrict ourselves
to the connected topological transformation groups in the base (which is the
situation that interests us most) then condition (3) holds for every
$G\in\mathsf{CAGp}$. Indeed, in such case all extensions
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ take their values in
$G_{0}$ and $\kappa_{k}(G_{0})=G_{0}$ for every $k\in\mathbb{N}$.

###### Proof of Theorem 4.12.

The equivalence of (1) and (2) follows immediately from the definition of
$\mathop{\rm\gimel_{\mathcal{F}}}$. Further, implications (5)$\Rightarrow$(4),
(3)$\Rightarrow$(2) and (4)$\Rightarrow$(2) hold obviously. Also, (5) implies
(3), since the endomorphisms $\kappa_{k}$ of $G$ ($k\in\mathbb{N}$) are
epimorphisms for every connected group $G\in\mathsf{CAGp}$. Thus, in order to
prove the equivalence of (1)–(5), it suffices now to show that (2) implies
(5). So assume that $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible. Fix
$G,H\in\mathsf{CAGp}$, an epimorphism $p\colon H\to G$ and
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. Then
$\mathcal{C}^{*}\in\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})$
and $p^{*}\colon G^{*}\to H^{*}$ is a monomorphism. By divisibility of
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, $\mathcal{C}^{*}$ extends through $p^{*}$
to a morphism $h\colon H^{*}\to\mathop{\rm{\bf Z}_{\mathcal{F}}}$; that is,
$hp^{*}=\mathcal{C}^{*}$. By virtue of Theorem 3.1, $h=\mathcal{D}^{*}$ for an
appropriate $\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(H)$. Then
$\mathcal{D}^{*}p^{*}=\mathcal{C}^{*}$, which translates into
$p\mathcal{D}=\mathcal{C}$. This verifies condition (5).

We finish the proof by verifying the equivalence (2)$\Leftrightarrow$(6).
First, assume that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible
and fix $G,H$ as in (6). Then $\mathop{\rm Ext}(H^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})=0$ and statement (6) thus follows from the exactness of
(3.3) in Corollary 3.4. Conversely, assume that condition (6) holds. We show
that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible by verifying
the identity $\mathop{\rm Ext}(A,\mathop{\rm{\bf Z}_{\mathcal{F}}})=0$ for
every $A\in\mathsf{DAGp}$. So fix $A\in\mathsf{DAGp}$ and set $H=A^{*}$.
Clearly, $H\mathrel{\lhd}G$ for an appropriate torus $G$. Since the group
$G^{*}$ is free abelian, it follows that $\mathop{\rm
Ext}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})=0$ and so the exact sequence
(3.3) takes the form

$0\longrightarrow\mathop{\rm{\bf
Z}_{\mathcal{F}}}(H)\stackrel{{\scriptstyle\widehat{j}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\widehat{p}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G/H)\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\mathop{\rm
Ext}(H^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})\longrightarrow 0.$ (4.1)

By virtue of condition (6), $\widehat{p}$ is an epimorphism and hence, by
exactness of the sequence (4.1), $\mathop{\rm Ext}(H^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})=0$. Thus, $\mathop{\rm Ext}(A,\mathop{\rm{\bf
Z}_{\mathcal{F}}})=0$, as was to be shown. ∎

######  Theorem 4.14.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow.
Then the following conditions are equivalent:

  1. (a)

$\mathop{\rm\gimel_{\mathcal{F}}}$ is torsion-free and connected,

  2. (b)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible and torsion-free,

  3. (c)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is divisible for every
$G\in\mathsf{CAGp}$,

  4. (d)

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ is divisible and torsion-free for every
$G\in\mathsf{CAGp}$,

  5. (e)

the objects of $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ lift uniquely across
the epimorphisms from $\mathsf{CAGp}$ with finite kernels,

  6. (f)

the objects of $\mathop{\rm\mathsf{CAGpZ}_{\mathcal{F}}}$ lift uniquely across
the epimorphisms from $\mathsf{CAGp}$ with totally disconnected kernels.

######  Remark 4.15.

In connection with condition (b), recall the following facts. If
$\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ is a minimal flow
whose acting group $\Gamma$ has no non-trivial finite abelian quotient groups
and whose base space $X$ is connected then the group $\mathop{\rm{\bf
Z}_{\mathcal{F}}}$ is torsion-free, see Theorem 4.10. If
$\Gamma\in\mathsf{CLAC}$ is simply connected and $X$ is compact then
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible, see Remark 3.39.

###### Proof of Theorem 4.14.

First, the equivalence of (a) and (b) follows immediately from the definition
of $\mathop{\rm\gimel_{\mathcal{F}}}$. Further, implications
(f)$\Rightarrow$(e) and (d)$\Rightarrow$(c) are obvious. Also, (b) follows
from (e) at once, for the endomorphisms $\kappa_{k}$ ($k\in\mathbb{N}$) of
$\mathbb{T}^{1}$ have finite kernels. To finish the proof, it is sufficient to
verify implications (b)$\Rightarrow$(d), (c)$\Rightarrow$(b) and
(b)$\Rightarrow$(f).

We show that (d) follows from (b). So assume that (b) holds and fix
$G\in\mathsf{CAGp}$. Since the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is
torsion-free, it follows that so is $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})$. Further, since $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is
torsion-free and divisible, all the morphisms $\kappa_{k}\colon\mathop{\rm{\bf
Z}_{\mathcal{F}}}\to\mathop{\rm{\bf Z}_{\mathcal{F}}}$ ($k\in\mathbb{N}$) are
isomorphisms and hence the group $\mathop{\rm Hom}(G^{*},\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ is divisible. Thus, $\mathop{\rm
Hom}(G^{*},\mathop{\rm{\bf Z}_{\mathcal{F}}})$ is both divisible and torsion-
free and so, by Theorem 3.1, it follows that so is $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$. This verifies condition (d).

We show that (b) follows from (c). So assume that all the groups
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ ($G\in\mathsf{CAGp}$) are divisible; we
show that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is divisible and
torsion-free. The divisibility of $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is
immediate. To see that $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is also torsion-
free, fix an integer $k\geq 2$. Then, by virtue of Theorem 3.1,
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{Z}_{k})\cong\mathop{\rm
Hom}(\mathbb{Z}_{k},\mathop{\rm{\bf Z}_{\mathcal{F}}})\cong\mathop{\rm
tor}_{k}(\mathop{\rm{\bf Z}_{\mathcal{F}}})$. Since $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{Z}_{k})$ is divisible by the assumptions, it follows
that $\mathop{\rm tor}_{k}(\mathop{\rm{\bf Z}_{\mathcal{F}}})=0$. This
verifies condition (b).

We finish the proof of the theorem by showing that (f) follows from (b). So
assume that $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is both divisible and torsion-
free. Fix an epimorphism $p\colon H\to G$ in $\mathsf{CAGp}$ with a totally
disconnected kernel and an extension $\mathcal{C}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)$; we wish to show that $p\mathcal{D}=\mathcal{C}$ holds
for a unique extension $\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(H)$.
The existence of such $\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(H)$
follows from the implication (2)$\Rightarrow$(5) from Theorem 4.12. To verify
the uniqueness part, assume that
$\mathcal{D}_{1},\mathcal{D}_{2}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(H)$ are
such that $p\mathcal{D}_{i}=\mathcal{C}$ for $i=1,2$. Then
$\mathcal{D}_{1}\mathcal{D}_{2}^{-1}\in\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\ker(p))$. Since $\ker(p)\in\mathsf{CAGp}$ is totally
disconnected, its dual group $\ker(p)^{*}$ is a torsion group and hence
$\chi(\mathcal{D}_{1}\mathcal{D}_{2}^{-1})\in\mathop{\rm tor}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})$ for every $\chi\in H^{*}$. By virtue of (b), the group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is torsion-free and hence
$\chi(\mathcal{D}_{1}\mathcal{D}_{2}^{-1})=1$ for every $\chi\in H^{*}$. Thus,
$\mathcal{D}_{1}=\mathcal{D}_{2}$, as was to be shown. ∎

######  Proposition 4.16.

Let $\mathop{\rm\mathcal{F}\colon\Gamma\curvearrowright X}$ be a minimal flow
and assume that the group $\mathop{\rm{\bf Z}_{\mathcal{F}}}$ is torsion-free
and divisible. Then the following statements hold.

  1. (i)

For every $G\in\mathsf{CAGp}$ and every $H\mathrel{\lhd}G$ totally
disconnected, the quotient morphism $p\colon G\to G/H$ induces an isomorphism
of groups

$\widehat{p}\colon\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)\to\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G/H).$

The isomorphism is topological, provided $\Gamma\in\mathsf{LieGp}$ is
connected and $X$ is compact.

  2. (ii)

For every $G\in\mathsf{CAGp}$ connected, the projective resolution _( 2.18)_
of $G$ induces isomorphisms of groups

$\left(\mathop{\rm{\bf Z}_{\mathcal{F}}}\right)^{\dim(G)}\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathfrak{P}(G))\stackrel{{\scriptstyle\widehat{\mathfrak{p}}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G).$

The isomorphisms are topological, provided $\Gamma\in\mathsf{LieGp}$ is
connected and $X$ is compact.

  3. (iii)

For every $G\in\mathsf{CAGp}$ connected, the maximal toral quotient sequence
_( 2.13)_ of $G$ induces isomorphisms of groups

$\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G)\stackrel{{\scriptstyle\widehat{q}}}{{\longrightarrow}}\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}(G))\cong(\mathop{\rm{\bf
Z}_{\mathcal{F}}})^{\dim(G)}.$

The isomorphisms are topological, provided $\Gamma\in\mathsf{LieGp}$ is
connected and $X$ is compact.

  4. (iv)

Assume that both $\Gamma$ and $X$ are locally compact second countable and
that $X$ is non-degenerate. Then there are an isomorphism of abelian groups
$\mathop{\rm{\bf Z}_{\mathcal{F}}}\cong\mathbb{R}$ and a topological
isomorphism $\mathop{\rm\gimel_{\mathcal{F}}}\cong b\mathbb{R}$.

###### Proof.

We begin by verifying statement (i). First, the morphism $\widehat{p}$ is an
isomorphism by virtue of statement (f) from Theorem 4.14. Further, let
$\Gamma\in\mathsf{LieGp}$ be connected and $X$ be compact. Then both
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$, $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(G/H)$ are topological groups with the topology of u.c.s.
convergence and $\widehat{p}$ is a bijective morphism of topological groups.
We show that $\widehat{p}$ is in fact a topological isomorphism by showing
that it is open at the identity. So let $\mathcal{U}$ be an identity
neighbourhood in $\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$. By our assumptions on
$\Gamma$ and $X$, there exist a compact connected set $1\in C\subseteq\Gamma$
and an identity neighbourhood $V$ in $G$ such that
$\mathcal{U}\supseteq[C\times X;V]$. Let $K\subseteq V$ be an open subgroup of
$H$. The quotient group $F=H/K$ is then finite. Further, for every $f\in F$,
fix $h_{f}\in f$. Since the cosets $K+h_{f}$ ($f\in F$) are compact, there is
an identity neighbourhood $V^{\prime}$ in $G$ with $V^{\prime}+K\subseteq V$
and with the sets $V^{\prime}+K+h_{f}$ ($f\in F$) mutually disjoint.

We claim that $\widehat{p}(\mathcal{U})\supseteq[C\times X;p(V^{\prime})]$;
since $[C\times X;p(V^{\prime})]$ is clearly an identity neighbourhood in
$\mathop{\rm{\bf Z}_{\mathcal{F}}}(G/H)$, this will finish the proof of (i).
So let $\mathcal{D}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G/H)$ be an extension
with $\mathcal{D}(C\times X)\subseteq p(V^{\prime})$ and let
$\mathcal{C}\in\mathop{\rm{\bf Z}_{\mathcal{F}}}(G)$ be the lift of
$\mathcal{D}$ across $p$; we show that $\mathcal{C}\in\mathcal{U}$. Then

$\mathcal{C}(C\times X)\subseteq
p^{-1}p(V^{\prime})=V^{\prime}+H=\bigcup_{f\in F}(V^{\prime}+K+h_{f}).$

Since the set $C\times X$ is connected, the sets $V^{\prime}+K+h_{f}$ ($f\in
F$) are open and mutually disjoint, and $e\in\mathcal{C}(C\times
X)\cap(V^{\prime}+K+h_{0})\neq\emptyset$, it follows that

$\mathcal{C}(C\times X)\subseteq V^{\prime}+K+h_{0}=V^{\prime}+K\subseteq V.$

Thus, $\mathcal{C}\in[C\times X;V]\subseteq\mathcal{U}$, as was to be shown.

We verify statement (ii). The fact that $\widehat{\mathfrak{p}}$ is a
(topological) isomorphism follows from statement (i), for the kernel
$\mathfrak{K}(G)$ of $\mathfrak{p}$ is totally disconnected. We verify the
other isomorphism from (ii). By definition of $\mathfrak{P}(G)$, there are
topological isomorphisms

$\mathfrak{P}(G)=(\mathbb{Q}\otimes
G^{*})^{*}\cong\left(\mathbb{Q}^{\left(\mathop{\rm
rank}(G^{*})\right)}\right)^{*}=\left(\mathbb{Q}^{\left(\dim(G)\right)}\right)^{*}\cong\left(\mathbb{Q}^{*}\right)^{\dim(G)}.$

Moreover, since $\mathfrak{P}(\mathbb{T}^{1})\cong\mathbb{Q}^{*}$, the part of
statement (ii) already proved yields an isomorphism $\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{Q}^{*})\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{T}^{1})=\mathop{\rm{\bf Z}_{\mathcal{F}}}$ (the
isomorphism is in fact topological, provided $\Gamma\in\mathsf{LieGp}$ is
connected and $X$ is compact). It follows that there are isomorphisms of
abelian groups

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathfrak{P}(G))\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}\left((\mathbb{Q}^{*})^{\dim(G)}\right)\cong\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}(\mathbb{Q}^{*})\right)^{\dim(G)}\cong\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}\right)^{\dim(G)}.$

If $\Gamma\in\mathsf{LieGp}$ is connected and $X$ is compact then all the
three isomorphisms above are topological. This finishes the proof of (ii).

We verify statement (iii). The fact that $\widehat{q}$ is a (topological)
isomorphism follows from statement (i), for the kernel $\mathbb{K}(G)$ of $q$
is totally disconnected. The other (topological) isomorphism from (iii)
follows from (topological) isomorphisms

$\mathop{\rm{\bf Z}_{\mathcal{F}}}(\mathbb{T}(G))\cong\mathop{\rm{\bf
Z}_{\mathcal{F}}}\left(\mathbb{T}^{\mathop{\rm
rank}(G^{*})}\right)=\mathop{\rm{\bf
Z}_{\mathcal{F}}}\left(\mathbb{T}^{\dim(G)}\right)\cong\left(\mathop{\rm{\bf
Z}_{\mathcal{F}}}\right)^{\dim(G)}.$

We finish the proof of the proposition by verifying statement (iv). The group
$\mathop{\rm{\bf Z}_{\mathcal{F}}}$, being divisible and torsion-free, is a
rational linear space. Write $\mathfrak{k}=\mathop{\rm rank}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})=\mathop{\rm dim_{\mathbb{Q}LS}}(\mathop{\rm{\bf
Z}_{\mathcal{F}}})$. By our assumptions on $\Gamma$ and $X$, $\mathop{\rm{\bf