# On locally compact groups of small topological entropy

Francesco G. Russo♮ Francesco G. Russo and Olwethu WakaDepartment of
Mathematics and Applied MathematicsUniversity of Cape TownPrivate Bag X1,
Rondebosch 7701Cape Town, South AfricaEmails<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Olwethu Waka

(Date: 4th of April 2023 ♮ Corresponding author)

###### Abstract.

We discuss the finiteness of the topological entropy of continuous
endomorphims for some classes of locally compact groups. Firstly, we focus on
the abelian case, imposing the condition of being compactly generated, and
note an interesting behaviour of slender groups. Secondly, we remove the
condition of being abelian and consider nilpotent periodic locally compact
$p$-groups ($p$ prime), reducing the computations to the case of Sylow
$p$-subgroups. Finally, we investigate locally compact Heisenberg $p$-groups
$\mathbb{H}_{n}(\mathbb{Q}_{p})$ on the field $\mathbb{Q}_{p}$ of the $p$-adic
rationals with $n$ arbitrary positive integer.

Keywords and Phrases: Topological Entropy; Locally Compact Groups; Dynamical
Systems; Sylow $p$-Subgroups; Slender Groups; Heisenberg $p$-Groups.

Mathematics Subject Classification 2020: 22A05, 37B40, 54C70.

##  1\. Motivations and Main Results

In the present paper a locally compact group is always assumed to be a
topological group whose topology is both Hausdorff and locally compact. Hood
[14] formulated a notion of topological entropy involving the well known
concept of uniformity for a topological space. His definition applies to a
topological groups possessing a left uniformity, since continuous
endomorphisms are uniformly continuous (in connection with the given left
uniformity). Let’s be more formal on Hood’s Entropy [14] in the context of
what we need to investigate here. For a locally compact group $G$, we denote
by $\mathcal{CT}(G)$ the collection of all compact neighborhoods of the
identity of $G$, and by $\mu$ a left invariant Haar measure on $G$. For a
continuous endomorphism $\varphi$ of $G$, an element $V\in\mathcal{CT}(G)$ and
an $n\in\mathbb{N}=\\{1,2,3,\ldots\\}$,

(1.1)
$C_{n}(\varphi,V)=V\cap\varphi^{-1}(V)\cap\ldots\cap\varphi^{-n+1}(V)\in\mathcal{CT}(G)$

defines the _$n$ -th $\varphi$-cotrajectory of $V$_. The topological entropy
of $\varphi$ (in the sense of Hood) is

(1.2)
$\mathsf{h}_{\mathsf{top}}(\varphi)=\sup\left\\{\limsup_{n\to\infty}\left(\frac{-\log\mu(C_{n}(\varphi,V))}{n}\right)\
\ \Big{|}\ \ V\in\mathcal{CT}(G)\right\\},$

Adler and others [1, 3, 18] investigated the aforementioned notions, stressing
on dynamical properties of topological structures with relations with ergodic
theory and mathematical physics.

Following [7, 8], we may introduce the _topological entropy of_ a locally
compact group $G$ as

(1.3)
$\mathsf{E}_{\mathsf{top}}(G)=\\{\mathsf{h}_{\mathsf{top}}(\varphi)\mid\varphi\in\mathrm{End}(G)\\},$

where $\mathrm{End}(G)$ denotes the ring of continuous endomorphisms of $G$
and $\mathrm{Aut}(G)$ the group of continuous automorphisms of $G$. Here we
investigate the cardinality of (1.3) and relations with structural properties,
as made in [3, 4, 8, 17, 21, 22, 24].

Denoting with $\widehat{\mathbb{Q}}$ the topological dual (in the sense of
Pontryagin) of the additive group $\mathbb{Q}$ of the rationals, we note that

(1.4) $\inf\\{\mathsf{E}_{\mathsf{top}}(G)\setminus\\{0\\}\mid G\ \text{is a
compact
group}\\}=\inf(\\{\mathsf{h}_{\mathsf{top}}(\varphi)\mid\varphi\in\mathrm{Aut}(\widehat{\mathbb{Q}}^{n}),\
n\in\mathbb{N}\\}\setminus\\{0\\})$

and a formula of Yuzvinski [24] shows that
$\mathsf{h}_{\mathsf{top}}(\varphi)$ can be calculated from the solutions of
the characteristic polynomial of $\varphi$ (see [16, 24]). Looking at locally
compact groups, we also note that $\mathsf{h}_{\mathsf{top}}(\psi)$ is finite
for any $\psi\in\mathrm{End}(\mathbb{R})$. Actually, we can do much more:
given $t\in\mathsf{E}_{\mathsf{top}}(\mathbb{R})\setminus\\{+\infty\\}$ we may
construct $\psi\in\mathrm{Aut}(\mathbb{R})$ of
$\mathsf{h}_{\mathsf{top}}(\psi)=t$, see [3, 23].

Following [7, 8, 10, 22], we introduce (for a locally compact group $G$)

(1.5)
$\mathfrak{E}_{0}=\\{G\mid\mathsf{E}_{\mathsf{top}}(G)=\\{0\\}\\}\quad\text{and}\quad\mathfrak{E}_{<\infty}=\\{G\mid\mathsf{E}_{\mathsf{top}}(G)=[0,+\infty)\\}$

and note that there are results, which describe the abelian cases in
$\mathfrak{E}_{<\infty}$ and $\mathfrak{E}_{0}$. The characterization of
groups in $\mathfrak{E}_{0}$ can indicate the presence of structural theorems.
For instance, finite abelian groups are in $\mathfrak{E}_{0}$ and have a
decomposition in direct product. On the other hand, very little is known in
the nonabelian case in $\mathfrak{E}_{<\infty}$ and $\mathfrak{E}_{0}$.

Following [12, Definition 2.2] and denoting by $\mathbb{P}$ the set of all
primes, an element $g$ of a locally compact group $G$ is called $p$-$element$,
if the sequence $g^{p^{k}}$ with $k\in\mathbb{N}$ tends to the identity
element in $G$. A locally compact group $G$ is called $p$-$group$, if $G$
coincides with

(1.6) $G_{p}=\\{g\in G\mid g\ \mbox{is a}\ p\mbox{-element}\\}=\\{g\in
G\mid\lim_{k\to\infty}g^{p^{k}}=1\\}.$

A maximal $p$-subgroup of a locally compact group $G$ is called $p$-$Sylow$
$subgroup$ of $G$. Note that $G_{p}$ turns out to be a closed subgroup by [12,
Lemma 2.6], when $G$ is totally disconnected. Following [12, 13], we denote by
$G_{0}$ the connected component of the identity and say that $G$ is compactly
covered, if for an arbitrary $x\in G$ we can always find a compact subgroup
$C$ of $G$ such that $x\in C$. From [12, p.5], a compact element of $G$ is an
element $g\in G$ such that $\overline{\langle g\rangle}$ is compact and the
set

(1.7) $\mathrm{comp}(G)=\\{g\in G\mid g\ \mbox{is a compact element }\\}$

is described in [12, Proposition 1.3, Lemma 1.6]. For instance,
$G=\mathrm{comp}(G)$ when $G$ is locally compact abelian, but in general
$\mathrm{comp}(G)$ is just a subset of $G$, not necessarily a subgroup. Note
that $\mathrm{comp}(G)$ is denoted by $B(G)$ in [2, 7, 8]; similarly, $G_{0}$
by $c(G)$. Following [12, Proposition 1.3], we call _periodic_ those locally
compact groups $G$ such that $G_{0}=1$ and $\overline{\langle g\rangle}$ is
compact for all $g\in G$. Of course, periodic locally compact groups are
totally disconnected, so their Sylow $p$-subgroups are closed and
$\mathrm{comp}(G)=G$ by [12, Lemma 1.6].

A locally compact group $G$ is topologically finitely generated, if there
exists a finite subset $X$ of $G$ such that $G=\overline{\langle X\rangle}$.
In particular, a locally compact $p$-group $G$ has ﬁnite p-rank, if

(1.8) $\mathrm{rank}_{p}(G)=\max\\{\mathrm{rank}_{p}(H)\ |\ H\ \mbox{closed
subgroup of}\ G\\}$

is a positive integer, where also the following quantities are positive
integers

(1.9) $\mathrm{rank}_{p}(H)=\min\\{|Y|\ |\ Y\subseteq H\ \mbox{and}\
\overline{\langle Y\rangle}=H\\}.$

For compact $p$-groups, see also [19, §2.4]. Following [12, 13], a locally
compact group $G$ is $compactly$ $generated$ if there exists a compact set $C$
such that $G=\langle C\rangle$. It is possible to provide examples of periodic
locally compact groups, which are not compactly generated. It is also possible
to provide examples which show that “topologically finitely generated groups”
and “compactly generated groups” are two different notions.

######  Theorem 1.1 (See, [13], Theorem 7.57).

Every compactly generated locally compact abelian group is isomorphic to a
direct sum $\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus K$ for a compact abelian
group $K$ and two nonnegative integers $d,m$.

We are going to focus on specific classes of locally compact abelian groups
and check whether the topological entropy of their continuous endomorphisms is
finite or not; results of the type of Theorem 1.1 are fundamental for this
scope. Denote the cartesian sum of countably many copies of $\mathbb{Z}$ by
$\mathbb{Z}^{\mathbb{N}}=\\{{(x_{i})}_{i\in\mathbb{N}}\ |\
x_{i}\in\mathbb{Z}\\}$ and by
$\mathbb{Z}^{(\mathbb{N})}=\\{{(x_{i})}_{i\in\mathbb{N}}\ |\
x_{i}\in\mathbb{Z}\ \mbox{and}\ x_{i}=0\ \mbox{for almost all}\ i\\}$ the
direct sum of countably many copies of $\mathbb{Z}$. Denote by
$\mathrm{dim}(A)$ the dimension of a compact abelian group $A$, that is, the
dimension of the $\mathbb{Q}$-module $\mathbb{Q}\otimes\widehat{A}$ as per
[13, Definitions 8.23]. Note also from [13, Corollary 7.58] that a connected
compact abelian group $A$ of finite dimension is characterized to be the
direct sum of finitely many copies of the torus
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$.

######  Definition 1.2 (See [9], p.489).

A (discrete) torsion-free abelian group $G$ is slender, if for every
homomorphism
$\alpha:{(e_{i})}_{i\in\mathbb{N}}\in\mathbb{Z}^{\mathbb{N}}\longmapsto\alpha({(e_{i})}_{i\in\mathbb{N}})\in
G$ we have $\alpha({(e_{i})}_{i\in\mathbb{N}})=0$ for almost all $i$, where
${(e_{i})}_{i\in\mathbb{N}}$ is the sequence with the $i$-th component equal
to 1 and 0 elsewhere.

Our first main result can be now formulated:

######  Theorem 1.3 (First Main Theorem).

Let $G$ be a compactly generated locally compact abelian group. With the
notations of Theorem 1.1, the following statements are satisfied :

  * (a).

If $G$ is slender, then $G\in\mathfrak{E}_{0}$. Viceversa, if
$G\in\mathfrak{E}_{0}$ and $K=0$, then $G$ is slender.

  * (b).

Assume that $K$ is connected. Then $G\in\mathfrak{E}_{<\infty}$ if and only if
$G\simeq\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus\mathbb{T}^{s}$ for some
nonnegative integers $d,m,s$.

Note that computations of the topological entropy of continuous automorphisms
(not endomorphisms) of
$\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus\mathbb{T}^{s}$ are available in [18,
pp. 475–476]. Also [4, 17] contain computations of the topological entropy of
continuous endomorphims, but mostly of Lie groups. We go ahead and describe
the finiteness of the topological entropy for some nonabelian locally compact
groups, looking at the behaviour of the Sylow $p$-subgroups. This is our
second main result.

######  Theorem 1.4 (Second Main Theorem).

The continuous automorphisms of a nilpotent periodic locally compact $p$-group
$G$ have finite topological entropy whenever $\mathrm{rank}_{p}(G)$ is finite.

We can always find periodic locally compact $p$-groups $G$ of
$\mathrm{rank}_{p}(G)=r$ in $\mathfrak{E}_{<\infty}$, looking at the direct
sum $G=\mathbb{Z}^{r}_{p}$ of $r$ copies of the additive group of $p$-adic
integers $\mathbb{Z}_{p}$. On the other hand, it is possible to find periodic
locally compact $p$-groups of nilpotency class two and of finite $p$-rank,
looking at Heisenberg $p$-groups $\mathbb{H}_{n}(\mathbb{Q}_{p})$ constructed
with upper triangular $(n+2)\times(n+2)$ matrices with coefficients in the
field of $p$-adic rationals $\mathbb{Q}_{p}$. These are neither abelian nor
compact groups, and have finite topological entropy and finite $p$-rank large
enough.

######  Theorem 1.5 (Third Main Theorem).

The Heisenberg group $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is a periodic locally
compact nonabelian $p$-group of nilpotency class $2$ with
$\mathrm{rank}_{p}(\mathbb{H}_{n}(\mathbb{Q}_{p}))=2n$, where $n$ is an
arbitrary positive integer. Moreover $\mathbb{H}_{n}(\mathbb{Q}_{p})$ belongs
to $\mathfrak{E}_{<\infty}$, but not to $\mathfrak{E}_{0}$.

Terminology and notations are standard and follow [9, 12, 13, 15, 19, 20].
After the statement of the main results in Section 1, the theory of slender
groups is summarized in Section 2 from [9, 20] and some recent results on the
finiteness of the topological entropy for periodic locally compact groups are
summarized in Section 3 from [1, 3, 10, 7, 8, 22]. Section 4 is devoted to
construct $\mathbb{H}_{n}(\mathbb{Q}_{p})$ and to prove some results on the
$p$-rank of these groups. Then we end with the proofs of Theorems 1.3, 1.4 and
1.5 in Section 5.

##  2\. Previous Results on Slender Groups

We recall properties of slender groups, originally noted by Nunke, Los and
Sasiada, see [9].

######  Lemma 2.1 (See [9], Chapter 13, §2).

  * (i).

Subgroups of slender groups are slender;

  * (ii).

Slender groups are torsion-free;

  * (iii).

$\mathbb{Q}$, $\mathbb{Z}_{p}$ and $\mathbb{Z}^{\mathbb{N}}$ are not slender;

  * (iv).

A group which is slender cannot contain any subgroup isomorphic to
$\mathbb{Q}$, $\mathbb{Z}_{p}$, or $\mathbb{Z}^{\mathbb{N}}$;

  * (v).

Direct products of slender groups are slender. In particular,
$\mathbb{Z}^{(\mathbb{N})}$ is slender;

  * (vi).

A torsion-free abelian group $G$ is slender if for every homomorpshim
$f:\mathbb{Z}^{\mathbb{N}}\to G$ the image $f(\mathbb{Z}^{\mathbb{N}})$ is a
discrete finitely generated abelian group.

From [12, 13, 15, 19], we may consider a periodic locally compact $p$-group
$G$ (not necessarily abelian) with $k$ positive integer and introduce the
subgroups

(2.1) $\Omega_{k}(G)=\overline{\langle g\mid g^{p^{k}}=1\rangle}\ \
\mbox{and}\ \ \Omega^{k}(G)=\langle g^{p^{k}}\mid g\in G\rangle,$

which are fully invariant in $G$ and satisfy $G/\Omega_{k}(G)=\Omega^{k}(G)$.
This allows us to introduce also

(2.2) $\mathrm{Div}(G)=\underset{k\in\mathbb{N}}{\bigcap}\Omega^{k}(G),$

which turns out to be useful for various reasons. For instance, if $G$ is an
(discrete) abelian group (not necessarily a periodic locally compact
$p$-group), $\mathrm{Div}(G)$ as above is still well defined and we say that
$G$ is $divisible$, if $\mathrm{Div}(G)\supseteq G$, or that $G$ is $reduced$,
if the trivial subgroup of $G$ is the only divisible subgroup of $G$ (see [13,
Appendix 1, Definition A1.29]).

######  Lemma 2.2 (See [13], Corollary 8.5).

For a compact abelian group $G$, the following conditions are equivalent:

  * (i).

$G$ is totally disconnected;

  * (ii).

$\mathrm{Div}(G)=0$;

  * (iii).

$\widehat{G}$ is a torsion group.

Nunke and Sasiada [9, Chapter 13, §2] showed that slender groups cannot be
divisible. The reader can refer also to [20, Exercise 4.4.9] and their result
is summarized below.

######  Lemma 2.3 (See [9], Lemma 2.3, Sasiada’s Theorem ).

An abelian group which is slender must be reduced. In addition, if the group
is countable, then the condition of being reduced is necessary and sufficient
to conclude that the group is slender.

Following the discussion in [9, Chapter 1, §7] and [6, §1], we may consider an
abelian group $G$ and a filter $\mathcal{F}$ in the subgroups lattice
$\mathrm{L}(G)$ of $G$. Automatically $\mathcal{F}$ defines a topology on $G$,
if we declare $\mathcal{B}=\\{U\mid U\in\mathcal{F}\\}$ to be a basis of open
neighborhoods at the identity of $G$ and if for every $g\in G$ the cosets
$g\mathcal{B}=\\{gU\mid U\in\mathcal{B}\\}$ form a basis of open neighborhoods
at $g\in G$. This topology is said to be a linear topology on $G$ (or more
precisely a linear $\mathcal{F}$-topology on $G$). Linear groups (in the sense
of Orsatti and De Marco) are abelian groups with linear topologies. A linear
group $G$ is complete, if it is Hausdorff and every Cauchy net in $G$ has a
limit in $G$. De Marco and Orsatti [6] studied Hausdorff linear groups:

######  Definition 2.4 (See [6]).

An abelian group $G$ belongs to the class $L\Omega$ if it admits a linear
complete and nondiscrete, Hausdorff topology. We say that $G$ belongs to the
class $L\Omega_{1}$, if it belongs to $L\Omega$ and in addition its topology
is metrizable.

In fact the conditions of Definition 2.4 are not verified simultaneously, that
is, there are abelian linear groups which are not Hausdorff, or abelian linear
groups which are not complete and so on. Of course, abelian groups in
$L\Omega_{1}$ are also in $L\Omega$, but examples can show that the viceversa
is false.

######  Theorem 2.5 (See [6], Theorem 2.3).

A torsion-free abelian group possesses a metrizable linear complete
nondiscrete topology if and only if it contains a copy of $\mathbb{Z}_{p},$ or
of $\mathbb{Z}^{\mathbb{N}}$ as subgroup.

Note that all groups of $L\Omega_{1}$ are classified by Theorem 2.5. Moreover
Lemma 2.1 shows that both $\mathbb{Z}_{p}$ and $\mathbb{Z}^{\mathbb{N}}$ are
not slender, hence Theorem 2.5 implies that $G$ cannot be slender, if it is
possible to endow $G$ of a metrizable linear complete nondiscrete topology.
This is reported below:

######  Theorem 2.6 (See [6], De Marco and Orsatti).

Let $G$ be a reduced torsion-free abelian group. Then $G$ is slender if and
only if $G$ does not belong to $L\Omega_{1}$.

Thanks to what we have seen until now :

######  Lemma 2.7.

There are no nontrivial compact abelian slender groups.

###### Proof.

Assume that $G$ is a compact abelian slender group. Lemma 2.3 along with Lemma
2.2 (a) and (b) imply that $G$ is totally disconnected. Then $G$ should be
profinite by [13, Theorem 1.34], hence projective limit of finite groups.
Profinite abelian groups are not slender; $\mathbb{Z}_{p}$ is a
counterexample. From the contradiction, there are no nontrivial compact
slender groups. ∎

##  3\. Previous Results for Periodic Locally Compact Groups

When we have a totally disconnected locally compact group $G$, van Dantzig [5]
proved that

(3.1) $\mathcal{U}(G)=\\{V\leq G\mid\text{$V$ compact and open}\\}$

is contained in $\mathcal{CT}(G)$ and is local basis. From [8, Proposition 3.4
], we have that

(3.2)
$\mathsf{h}_{\mathsf{top}}(\varphi)=\sup\Big{\\{}\lim_{n\to\infty}\left(\frac{\log|V:C_{n}(\varphi,V)|}{n}\right)\
\ \Big{|}\ \ V\in\mathcal{U}(G)\Big{\\}},$

where $C_{n}(\varphi,V)\in\mathcal{U}(G)$ and the index $|V:C_{n}(\varphi,V)|$
is finite. In fact, the set $\mathsf{E}_{\mathsf{top}}(G)$ turns out to be a
countable subset of the real half-line in this situation.

Some relevant facts are reported below. The first regards discrete groups.

######  Remark 3.1 (See [8], Remark 2.4).

Discrete groups belong to $\mathfrak{E}_{0}$.

The second regards the additive group of $p$-adic integers.

######  Corollary 3.2 (See [8], Corollary 2.2).

Let $G$ be a locally compact group and $\varphi\in\mathrm{End}(G)$. If
$\mathcal{S}\subseteq\mathcal{CT}(G)$ is a local basis of $G$ and
$\mathcal{S}$ is realized by $\varphi$-invariant subgroups, then
$\mathsf{h}_{\mathsf{top}}(\varphi)=0$. In particular, this applies to
$\mathbb{Z}_{p}^{n}$, hence $\mathbb{Z}_{p}^{n}\in\mathfrak{E}_{0}$.

The computation of the topological entropy of continuous endomorphisms is
somehow harder than that of continuous automorphisms, but we have results for
totally disconnected groups.

######  Corollary 3.3 (See [8], Lemma 2.3, Theorem 3.11; See [10], Corollary
1.3).

Let $G$ be a locally compact group and $\varphi\in\mathrm{End}(G)$.

  1. (a).

If $H$ is a $\varphi$-invariant closed subgroup of $G$, then
$\mathsf{h}_{\mathsf{top}}({\varphi_{|}}_{H})\leq\mathsf{h}_{\mathsf{top}}(\varphi)$,
and, if in addition $H$ is normal, then
$\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{G/H})\leq\mathsf{h}_{\mathsf{top}}(\varphi)$,
where $\bar{\varphi}_{G/H}:G/H\to G/H$ is induced by $\varphi$.

  2. (b).

If $\mathcal{S}\subseteq\mathcal{U}(G)$ is a local basis of $G$ such that
$\varphi^{-n}(V)$ is normal in $G$ for all $n$ and $V\in\mathcal{S}$, then
$\mathsf{h}_{\mathsf{top}}(\varphi)=\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{G/\ker\varphi})$.

  3. (c).

If $G$ is totally disconnected and $\varphi\in\mathrm{Aut}(G)$, then
$\mathsf{h}_{\mathsf{top}}(\varphi)=\mathsf{h}_{\mathsf{top}}(\varphi_{|N})+\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{G/N})$,
where $N$ is a closed normal subgroup of $G$.

The third regards $p$-adic rationals. Denoting the $p$-adic norm with
$|-|_{p}$, Yuzvinski’s Formula [16, 24] helps with the following computations:

######  Theorem 3.4 (See [16]).

For $n\in\mathbb{N}$ and $\varphi\in\mathrm{End}(\mathbb{Q}_{p}^{n})$, we have

(3.3)
$\mathsf{h}_{\mathsf{top}}(\varphi)=\sum_{|\lambda_{i}|_{p}>1}\log|\lambda_{i}|_{p},$

where $\lambda_{i}$ (with $1\leq i\leq n$) is eigenvalue of $\varphi$ in a
finite extension of $\mathbb{Q}_{p}$. In particular,
$\mathbb{Q}_{p}^{n}\in\mathfrak{E}_{<\infty}$.

Further criteria of finiteness are related to the notion of of finite
$p$-rank.

######  Theorem 3.5 (See [12], Theorem 3.97).

A locally compact abelian $p$-group $G$ has finite $p$-rank if and only
$G\simeq\mathbb{Z}_{p}^{\alpha}\times\mathbb{Q}_{p}^{\beta}\times\mathbb{Z}(p^{\infty})^{\gamma}\times
E_{p}$ for some nonnegative integers $\alpha,\beta,\gamma,\delta$ and a finite
$p$-group $E_{p}$ of $\mathrm{rank}_{p}(E_{p})=\delta$. In particular, $G$
belongs to $\mathfrak{E}_{<\infty}$ and
$\mathrm{rank}_{p}(G)=\alpha+\beta+\gamma+\delta.$ The case of $G$ in
$\mathfrak{E}_{0}$ is characterized by the condition $\beta=0$.

The above result shows that the $p$-rank is preserved under Pontryagin
duality. Indeed,

(3.4)
$\widehat{G}={(\mathbb{Z}_{p}^{\alpha}\times\mathbb{Q}_{p}^{\beta}\times\mathbb{Z}(p^{\infty})^{\gamma}\times
F_{p})}^{\wedge}\cong\mathbb{Z}_{p}^{\gamma}\times\mathbb{Q}_{p}^{\beta}\times\mathbb{Z}(p^{\infty})^{\alpha}\times
F_{p},$

and so $\mathrm{rank}_{p}(\widehat{G})=\mathrm{rank}_{p}(G)$. In particular,
it can be seen that
$\mathbb{Q}_{p}^{\beta}\in\mathfrak{E}_{<\infty}\setminus\mathfrak{E}_{0}$,
$\mathbb{R}^{d}\in\mathfrak{E}_{<\infty}\setminus\mathfrak{E}_{0}$,
$\mathbb{Z}^{\gamma}_{p}\in\mathfrak{E}_{0}$, $F_{p}\in\mathfrak{E}_{0}$ and
${\mathbb{Z}(p^{\infty})}^{\alpha}\in\mathfrak{E}_{0}$. Further results are
reported below in the abelian case.

######  Theorem 3.6 (See [8], Theorems 1.1 and 1.2).

Let $G$ be a locally compact abelian group.

  * (i).

If $G$ belongs to $\mathfrak{E}_{<\infty}$, then its dimension should be
finite;

  * (ii).

The viceversa of (i) above is true when $G$ is compact and $G/G_{0}$ belongs
to $\mathfrak{E}_{<\infty}$;

  * (iii).

If $G$ belongs to $\mathfrak{E}_{0}$, then $G$ is totally disconnected;
moreover a profinite group belongs to $\mathfrak{E}_{0}$ if and only if it
belongs to $\mathfrak{E}_{<\infty}$;

  * (iv).

If $G$ is periodic, then $G$ is in $\mathfrak{E}_{0}$ if and only if all its
$p$-Sylow subgroups $G_{p}$ do the same.

In the arguments which are used to prove Theorem 3.6, the main logic is to
find decompositions of the endomorphisms in portions where we can control the
finiteness of the topological entropy. In fact we say that _the Addition
Theorem holds_ for $(G,\varphi,H)$ of a locally compact group $G$ with
$\varphi\in\mathrm{End}(G)$ and a $\varphi$-invariant closed normal subgroup
$N$ of $G$, if

(3.5)
$\mathsf{h}_{\mathsf{top}}(\varphi)=\mathsf{h}_{\mathsf{top}}({\varphi_{|}}_{N})+\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{G/N}),$

or briefly, we write that $AT(G,\varphi,N)$ _holds_. Of course, (3.5) is
equivalent to the commutativity of the following diagram

(3.6) $\begin{CD}0@>{}>{}>N@>{\iota}>{}>G@>{\pi}>{}>G/N@>{}>{}>0\\\
@V{{\varphi_{|}}_{N}}V{}V@V{\varphi}V{}V@V{\bar{\varphi}_{G/N}}V{}V\\\
0@>{}>{}>N@>{\iota}>{}>G@>{\pi}>{}>G/N@>{}>{}>0\\\ \end{CD}$

Similarly, _$AT(G)$ holds_ if $AT(G,\varphi,N)$, which is depending on
$\varphi$ and $N$ in general, is satisfied by all $\varphi$ and $N$. From [8,
Proposition 3.6], if $N$ is a fully invariant open subgroup of $G$ and $AT(N)$
holds, then also $AT(G)$ holds.

At this point it is important that we pause and look closely at the structure
of compactly generated locally compact abelian groups of Theorem 1.1. First,
we note that the groups that appear in the decomposition are either compact or
totally disconnected, or isomorphic to $\mathbb{R}^{d}$ for some nonnegative
integer $d$. Because of this observation, we record the following result:

######  Lemma 3.7 (See [8], Lemma 3.1).

Let $A$, $B$ be two locally compact groups that either are compact, or totally
disconnected or isomorphic to $\mathbb{R}^{d}$ for some nonnegative integer
$d$, and $f\in\mathrm{End}(A),$ $g\in\mathrm{End}(B)$. Consider $A\times B$
with the product topology and $f\times g\in\mathrm{End}(A\times B)$. Then

(3.7) $\mathsf{h}_{\mathsf{top}}(f\times
g)=\mathsf{h}_{\mathsf{top}}(f)+\mathsf{h}_{\mathsf{top}}(g).$

Again the situation is very clear computationally for locally compact abelian
groups.

######  Theorem 3.8 (See [8], Theorems 1.8 and 1.9).

Let $G$ be a totally disconnected locally compact abelian group. Then, for
every $\varphi\in\mathrm{End}(G)$, we have

(3.8)
$\mathsf{h}_{\mathsf{top}}(\varphi)=\sum_{p\in\mathbb{P}}\mathsf{h}_{\mathsf{top}}({\varphi_{|}}_{G_{p}}).$

If $G$ is also periodic, then $AT(G)$ holds if and only if $AT(G_{p})$ holds
for all $p$-Sylow subgroups $G_{p}$.

Theorem 3.8 (ii) shows that Addition Theorems may be reduced to Addition
Theorems on $p$-Sylow subgroups. This means that the presence of a
decomposition helps to determine groups in $\mathfrak{E}_{0}$ or in
$\mathfrak{E}_{<\infty}$, just looking at Sylow $p$-subgroups in
$\mathfrak{E}_{0}$ or in $\mathfrak{E}_{<\infty}$.

######  Remark 3.9.

For compactly generated locally compact abelian groups, Theorem 1.1 shows that
Lemma 3.7 can be applied and so we have an Addition Theorem. This helps to
reduce the computation of the topological entropy of continuous endomorphisms
to the topological entropy of continuous endomorphisms arising from factors.

##  4\. Heisenberg groups on p-adic rationals

As application of Corollary 3.2, we have that a compact $p$-group $G$ with
local basis $\\{\Omega^{n}(G)\mid n\in\mathbb{N}\\}\subseteq\mathcal{U}(G)$
should belong to $\mathfrak{E}_{0}$. Note that this applies to
$\mathbb{Z}_{p}^{n}\times F_{p}$, where $F_{p}$ is finite $p$-group.

######  Remark 4.1.

Groups of the form $\mathbb{Z}_{p}\times F_{p}$ for $F_{p}$ finite nonabelian
$p$-group are among the easiest examples of infinite nilpotent compact
$p$-groups which can be produced in $\mathfrak{E}_{0}$. Looking at [15,
Section 3.1], a finite $p$-group $F_{p}$ is of maximal class if $|p^{n}|$ with
$n>3$ and its nilpotency class is $c=n-1$. Their costruction can be found in
[15, Examples 3.1.5]. Now $\mathbb{Z}_{p}\times F_{p}$ has nilpotency class
exactly $n$ by Fitting’s Lemma [15, Lemma 1.1.21]. This means that we have
already an example of an infinite nonabelian compact $p$-group of nilpotency
class arbitrarily large in $\mathfrak{E}_{0}$.

Given a commutative unitary topological ring $R$, the Heisenberg group on $R$
is the group of all $(n+2)\times(n+2)$-matrices of the following form

(4.1) $M(A,B;c)=\left(\begin{array}[]{c|cccc|c}1&a_{1}&a_{2}&...&a_{n}&c\\\
\hline\cr 0&1&0&...&0&b_{1}\\\ 0&0&1&...&0&b_{2}\\\ ...&...&...&...&...&...\\\
0&0&0&...&1&b_{n}\\\ \hline\cr 0&0&0&0&0&1\\\
\end{array}\right)=\left(\begin{array}[]{cccc}1&A&c\\\ O&I_{n}&B\\\ 0&O&1\\\
\end{array}\right),$

where the block $O$ is of all zeros, $I_{n}$ denotes a identity matrix
$n\times n$, $A$ the $n$-tuple row $(a_{1},\ldots,a_{n})$, $B$ the $n$-tuple
column $(b_{1},\ldots,b_{n})$. Of course, for $n=1$ we get the usual
representation of the Heisenberg group as group of matrices $3\times 3$.

In particular, the matrices (4.1) have coefficients $m_{ij}$ such that

(4.2) $m_{ij}=\left\\{\begin{array}[]{lcl}1,\ \mbox{if}\ i=j,\\\ 0,\
\mbox{if}\ i>j,\ \mbox{or}\ 1<i<j<n-1.\\\ \end{array}\right.$

Note that $\mathrm{GL}(R^{n+2})$ is the general linear group of dimension
$n+2$ of all invertible matrices with coefficients in $R$, and the set of all
matrices (4.1) is denoted by $\mathbb{H}_{n}(R)$ and equipped with the product
topology induced by the product topology in $R^{{(n+2)}^{2}}$. In particular,
one can check that $\mathbb{H}_{n}(R)$ is nilpotent of class $2$, since the
center

(4.3)
$Z(\mathbb{H}_{n}(R))=\overline{[\mathbb{H}_{n}(R),\mathbb{H}_{n}(R)]}=\left\\{\left(\begin{array}[]{cccc}1&O&c\\\
O&I_{n}&O\\\ 0&O&1\\\ \end{array}\right)\ \mid\ \ c\in R\right\\}$

is topologically isomorphic to $(R,+)$ and the central quotient

(4.4)
$\mathbb{H}_{n}(R)/Z(\mathbb{H}_{n}(R))\cong\underbrace{(R,+)\times(R,+)\times\ldots\times(R,+)}_{2n-\mbox{times}},$

is topologically isomorphic to $2n$ copies of $(R,+)$. Note that for
$R=\mathbb{Z}$, or $\mathbb{Z}_{p}$, or $\mathbb{Z}(p)$,(4.4) is topologically
generated by the matrices of the following form

(4.5) $\left(\begin{array}[]{c|cccc|c}1&1&0&...&0&1\\\ \hline\cr
0&1&0&...&0&0\\\ 0&0&1&...&0&0\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&0\\\
\hline\cr 0&0&0&0&0&1\\\ \end{array}\right),\ \
\left(\begin{array}[]{c|cccc|c}1&0&1&...&0&1\\\ \hline\cr 0&1&0&...&0&0\\\
0&0&1&...&0&0\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&0\\\ \hline\cr
0&0&0&0&0&1\\\ \end{array}\right),\ \ \ldots\\\
\left(\begin{array}[]{c|cccc|c}1&0&0&...&1&1\\\ \hline\cr 0&1&0&...&0&0\\\
0&0&1&...&0&0\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&0\\\ \hline\cr
0&0&0&0&0&1\\\ \end{array}\right),$

along with the corresponding ones where the role of $A$ is played by $B$ in
the last column

(4.6) $\left(\begin{array}[]{c|cccc|c}1&0&0&...&0&1\\\ \hline\cr
0&1&0&...&0&1\\\ 0&0&1&...&0&0\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&0\\\
\hline\cr 0&0&0&0&0&1\\\ \end{array}\right),\ \
\left(\begin{array}[]{c|cccc|c}1&0&0&...&0&1\\\ \hline\cr 0&1&0&...&0&0\\\
0&0&1&...&0&1\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&0\\\ \hline\cr
0&0&0&0&0&1\\\ \end{array}\right),\ \ \ldots\\\
\left(\begin{array}[]{c|cccc|c}1&0&0&...&0&1\\\ \hline\cr 0&1&0&...&0&0\\\
0&0&1&...&0&0\\\ ...&...&...&...&...&...\\\ 0&0&0&...&1&1\\\ \hline\cr
0&0&0&0&0&1\\\ \end{array}\right).$

In particular, we describe the nonabelian compact $p$-group
$\mathbb{H}(\mathbb{Z}_{p})$ below for $n=1$.

Note from [19, Chapter 2] that the Frattini subgroup $\mathrm{Frat}(G)$ of a
profinite group $G$ is defined as the intersection of all its maximal open
subgroups. Moreover it is a characteristic subgroup of $G$. An element $g$ of
a profinite group $G$ is a nongenerator if it can be omitted from every
generating set of $G$, that is, whenever $G=\overline{\langle X,g\rangle}$,
then $G=\overline{\langle X\rangle}$. In particular,

######  Remark 4.2.

We have that the set of all nongenerators of a profinite group $G$ coincides
with $\mathrm{Frat}(G)$, see [19, Lemma 2.8.1]. Moreover [19, Lemma 2.8.6]
shows that the minimal number of generators of a topologically finitely
generated profinite group $G$ agrees with the minimal number of generators of
$G/\mathrm{Frat}(G)$. For compact $p$-group $G$, this means that
$\mathrm{rank}_{p}(G)=\mathrm{rank}_{p}(G/\mathrm{Frat}(G))$.

######  Example 4.3.

For any prime $p$, consider a separated bilinear map

(4.7)
$\omega:(x,y)\in\mathbb{Z}_{p}\times\mathbb{Z}_{p}\to\omega(x,y)\in\mathbb{Z}_{p}$

and the set $\mathbb{Z}_{p}^{3}$ endowed with the binary operation

(4.8) $\square\ :\
((x_{1},y_{1},z_{2}),(x_{2},y_{2},z_{2}))\in\mathbb{Z}_{p}^{3}\times\mathbb{Z}_{p}^{3}\mapsto(x_{1},y_{1},z_{1})\
\square\ (x_{2},y_{2},z_{2})$ $=\ \Big{(}x_{1}+x_{2},\ \ y_{1}+y_{2},\ \
z_{1}+z_{2}+\omega(x_{1},y_{2})\Big{)}\in\mathbb{Z}_{p}^{3}.$

See terminology in [2, Definitions 2.1 and 2.2]. In particular, $\square$ is
not a commutative operation and $(\mathbb{Z}_{p}^{3},\square)$ satisfies the
algebraic axioms of group. Of course, the construction depends on $\omega$ and
$(\mathbb{Z}_{p}^{3},\square)$ is topologically isomorphic to
$\mathbb{H}(\mathbb{Z}_{p})$ with the matrix product. The center, the Frattini
subgroup and the derived subgroup of $\mathbb{H}(\mathbb{Z}_{p})$ satisfy

(4.9) $\mathrm{Frat}(\mathbb{H}(\mathbb{Z}_{p}))\supseteq
Z(\mathbb{H}(\mathbb{Z}_{p}))=\\{M(0,0;c)\ |\
c\in\mathbb{Z}_{p}\\}=\overline{[\mathbb{H}(\mathbb{Z}_{p}),\mathbb{H}(\mathbb{Z}_{p})]}\simeq\mathbb{Z}_{p}.$

We can now look at topological generators and relations for
$\mathbb{H}(\mathbb{Z}_{p})$, finding that

(4.10) $\mathbb{H}(\mathbb{Z}_{p})=\overline{\langle
M(1,0;0),M(0,1;0),M(0,0;1)\ |\ [M(1,0;0),M(0,1;0)]=M(0,0;1),}$
$\overline{[M(1,0;0),M(0,0;1)]=[M(0,1;0),M(0,0;1)]=I_{3}\rangle}.$

From Remark 4.2, the $p$-rank of $\mathbb{H}(\mathbb{Z}_{p})$ can be reduced
to the Frattini quotient, i.e.:

(4.11)
$\mathrm{rank}_{p}(\mathbb{H}(\mathbb{Z}_{p}))=\mathrm{rank}_{p}(\mathbb{H}(\mathbb{Z}_{p})/\mathrm{Frat}(\mathbb{H}(\mathbb{Z}_{p})))=2.$

Indeed, $\mathrm{Frat}(G)=\overline{\Omega^{1}(G)[G,G]}$ for any compact
$p$-group $G$ by [19, Lemma 2.8.7 (c)]. Since this is true of course when
$G=\mathbb{H}(\mathbb{Z}_{p})$, in (4.9) one can compute
$\mathrm{Frat}(\mathbb{H}(\mathbb{Z}_{p}))$ as follows. Now
$[\mathbb{H}(\mathbb{Z}_{p}),\mathbb{H}(\mathbb{Z}_{p})]=Z(\mathbb{H}(\mathbb{Z}_{p}))$
and

(4.12)
$\Omega^{1}(\mathbb{H}(\mathbb{Z}_{p}))=\left(\begin{array}[]{cccc}1&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
0&1&p\mathbb{Z}_{p}\\\ 0&0&1\\\ \end{array}\right)$

are compact, so
$\mathrm{Frat}(\mathbb{H}(\mathbb{Z}_{p}))=\Omega^{1}(\mathbb{H}(\mathbb{Z}_{p}))Z(\mathbb{H}(\mathbb{Z}_{p}))$,
since the subgroup
$\Omega^{1}(\mathbb{H}(\mathbb{Z}_{p}))Z(\mathbb{H}(\mathbb{Z}_{p}))$ is
compact. Therefore,
$\mathbb{H}(\mathbb{Z}_{p})/\mathrm{Frat}(\mathbb{H}(\mathbb{Z}_{p}))\cong\mathbb{Z}(p)\times\mathbb{Z}(p)$.
This proves the second equality on (4.11). Example 4.3 holds more generally
than $R=\mathbb{Z}_{p}$, see [11, Theorem 2.5 and Lemma 5.5] and [2, §4]. Now
we look at $\mathbb{H}_{n}(\mathbb{Z}_{p})$ and
$\mathbb{H}_{n}(\mathbb{Q}_{p})$ for $n$ large enough.

######  Lemma 4.4.

The Heisenberg group $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is a locally compact
nonabelian $p$-group of nilpotency class two and

$\mathrm{rank}_{p}(\mathbb{H}_{n}(\mathbb{Q}_{p}))=\mathrm{rank}_{p}(\mathbb{H}_{n}(\mathbb{Q}_{p})/Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))=2n.$

###### Proof.

Looking at (4.1), (4.2), (4.3) and (4.4) with $R=\mathbb{Q}_{p}$, it is clear
that $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is a locally compact nonabelian
$p$-group of nilpotency class two. Now consider (4.1) and observe that

(4.13) $H_{1}=Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))\times
K_{2}\simeq\mathbb{Q}_{p}\times\mathbb{Q}^{n}_{p}\ \ \mbox{and}\ \
K_{1}\simeq\mathbb{Q}^{n}_{p}.$

Moreover $H_{1}$ is a closed normal subgroup such that

(4.14) $\mathbb{H}_{n}(\mathbb{Q}_{p})=H_{1}\rtimes K_{1}=\\{h_{1}k_{1}\mid
h_{1}\in H_{1}\ \mbox{and}\ k_{1}\in K_{1}\\}$ $=\\{zk_{2}k_{1}\mid z\in
Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))\ k_{2}\in K_{2},k_{1}\in
K_{1}\\}=\\{[u_{2},u_{1}]k_{2}k_{1}\mid k_{2},u_{2}\in K_{2}\ \mbox{and}\
k_{1},u_{1}\in K_{1}\\},$

because we have

(4.15)
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))=\overline{[\mathbb{H}_{n}(\mathbb{Q}_{p}),\mathbb{H}_{n}(\mathbb{Q}_{p})]}=\overline{[K_{2},K_{1}]}.$

Therefore the $p$-rank of $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is reduced to that
of $K_{1}$ plus that of $K_{2}$, i.e.: $2n$. ∎

Lemma 4.4 can be proved with the idea of Example 4.3, that is, noting that

(4.16) $\mathrm{Frat}(\mathbb{H}_{n}(\mathbb{Q}_{p}))\supseteq
Z(\mathbb{H}_{n}(\mathbb{Q}_{p})),$

and that the quotient
$\mathbb{H}_{n}(\mathbb{Q}_{p})/\mathrm{Frat}(\mathbb{H}_{n}(\mathbb{Q}_{p}))$
has $p$-rank $2n$, but we gave an argument based on the structure of
semidirect product for Heisenberg groups. Moreover also here one could argue
that
$\mathrm{Frat}(\mathbb{H}_{n}(\mathbb{Q}_{p}))=\overline{\Omega^{1}(\mathbb{H}_{n}(\mathbb{Q}_{p}))[\mathbb{H}_{n}(\mathbb{Q}_{p}),\mathbb{H}_{n}(\mathbb{Q}_{p})]}$,
even if in this sitation we don’t have a compact $p$-group but a periodic
locally compact $p$-group.

######  Remark 4.5.

Looking at [2, Lemma 2.4 and Theorem 2.5], one can show that
$\mathbb{H}(\mathbb{Z}_{p})$ possesses abelian maximal subgroups of the
following form

(4.17) $H_{1}=Z(\mathbb{H}(\mathbb{Z}_{p}))\oplus\overline{\langle
M(1,0;0)\rangle}\simeq{\mathbb{Z}_{p}}^{2}\ \ \mbox{and}\ \
H_{2}=Z(\mathbb{H}(\mathbb{Z}_{p}))\oplus\overline{\langle
M(0,1;0)\rangle}\simeq{\mathbb{Z}_{p}}^{2}$

satisfying the following conditions:

(4.18) $H_{1}\cap H_{2}=Z(\mathbb{H}(\mathbb{Z}_{p})),\ \
H_{1}\cap\overline{\langle M(0,1;0)\rangle}=1,\ \ H_{2}\cap\overline{\langle
M(1,0;0)\rangle}=1,$ (4.19)
$\mathbb{H}(\mathbb{Z}_{p})=H_{1}\rtimes\overline{\langle
M(0,1;0)}\rangle=H_{2}\rtimes\overline{\langle
M(1,0;0)\rangle}\simeq{\mathbb{Z}_{p}}^{2}\rtimes\mathbb{Z}_{p}.$

In particular (4.19) shows that any element of $\mathbb{H}(\mathbb{Z}_{p})$
can be written uniquely as product of an element of $H_{1}$ and of one of
$\overline{\langle M(0,1;0)\rangle}=K_{1}$, but any element of $H_{1}$ can be
also written uniquely as product of an element of
$Z(\mathbb{H}(\mathbb{Z}_{p}))$ and of one of $\overline{\langle
M(1,0;0)\rangle}=K_{2}$ by (4.17). In Fig. 1 we identify the aforementioned
subgroups in the lattice of closed subgroups
$\mathcal{SUB}(\mathbb{H}(\mathbb{Z}_{p}))$ of $\mathbb{H}(\mathbb{Z}_{p})$.
At the first level (beginning from the bottom of Fig. 1) we find the trivial
subgroup. At the second level there are three subgroups isomorphic to
$\mathbb{Z}_{p}$. At the third level there are two subgroups isomorphic to the
additive group ${\mathbb{Z}_{p}}^{2}$. At the fourth level we find the entire
group. Note that Fig. 1 shows only the subgroups that can be directly deduced
from (4.19) and not all the subgroups of $\mathbb{H}(\mathbb{Z}_{p})$.

$\mathbb{H}(\mathbb{Z}_{p})$$H_{1}$$K_{2}$$H_{2}$$K_{1}$$Z(\mathbb{H}(\mathbb{Z}_{p}))$$1$

Figure 1: Some relevant subgroups in
$\mathcal{SUB}(\mathbb{H}(\mathbb{Z}_{p}))$.

In fact one can see that, given the cardinality of the continuum
$\mathfrak{c}$ and fixed $\xi\in\mathbb{Z}_{p}$, the subset $M_{\xi}$ of all
matrices $M(a,\xi a;t)\in\mathbb{H}(\mathbb{Z}_{p})$ is a maximal abelian
subgroup of $\mathbb{H}(\mathbb{Z}_{p})$, and of course there are
$\mathfrak{c}$ of this type.

##  5\. Proofs of Main Theorems

###### Proof of Theorem 1.3.

(a). If $G$ is a compactly generated locally compact abelian group, then
Theorem 1.1 implies $G\cong\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus K$ for a
compact abelian group $K$ and nonnegative integers $m,d$. Assume in addition
that $G$ is slender. Lemma 2.1 shows that subgroups of slender groups are
slender. Then Lemma 2.3 implies $n=0$, that is, $G\simeq\mathbb{Z}^{m}\oplus
K$. Lemma 2.7 implies $K=0$. Hence $G\simeq\mathbb{Z}^{m}$, and since
$\mathbb{Z}^{m}\in\mathfrak{E}_{0}$, the first part of the result follows.
Assume now that $G\in\mathfrak{E}_{0}$ and that
$G\cong\mathbb{R}^{d}\oplus\mathbb{Z}^{m}$. Since
$\mathbb{R}^{d}\in\mathfrak{E}_{\infty}\setminus\mathfrak{E}_{0}$, $G$ should
be totally disconnected by Theorem 3.6 (iii) and so $G\simeq\mathbb{Z}^{m}$
which is slender. The result follows completely.

(b). From Theorem 1.1 and the assumption that $K$ is a connected compact
abelian group, we have that $G\simeq\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus
K$ with $K$ of $\mathrm{dim}(K)$ eventually infinite. Then

(5.1)
$\mathrm{dim}(G)=\mathrm{dim}(\mathbb{R}^{d})+\mathrm{dim}(\mathbb{Z}^{m})+\mathrm{dim}(K)=d+0+\mathrm{dim}(K)$

and this shows that $\mathrm{dim}(G)<\infty$ if and only if
$\mathrm{dim}(K)<\infty$ if and only if $K=\mathbb{T}^{s}$ for some
nonnegative integer $s$, see [13, Corollary 8.22 (5)]. From Theorem 3.6 (i),
this means that if $G\in\mathfrak{E}_{<\infty}$, then $\mathrm{dim}(G)<\infty$
hence $\mathrm{dim}(K)<\infty$, and so
$G\simeq\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus\mathbb{T}^{s}$. Conversely,
assume that $G\simeq\mathbb{R}^{d}\oplus\mathbb{Z}^{m}\oplus\mathbb{T}^{s}$.
We may apply Lemma 3.7 with summands
$\mathbb{R}^{d}\in\mathfrak{E}_{<\infty}$, $\mathbb{Z}^{m}\in\mathfrak{E}_{0}$
and $\mathbb{T}^{s}\in\mathfrak{E}_{<\infty}$, concluding
$G\in\mathfrak{E}_{<\infty}$. Note that the computations of topological
entropy, which allows us to have $\mathbb{R}^{d}\in\mathfrak{E}_{<\infty}$,
$\mathbb{Z}^{m}\in\mathfrak{E}_{0}$ and
$\mathbb{T}^{s}\in\mathfrak{E}_{<\infty}$, are well known, see [3, 16, 22].
The result follows.

∎

###### Proof of Theorem 1.4.

First assume that $G$ has $\mathrm{rank}_{p}(G)<\infty$. We note that closed
subgroups and quotients of $G$ are again periodic locally compact $p$-groups.
The topological lower central series of $G$ of length $c$ has closed
characteristic $p$-subgroups $\overline{\gamma_{i}(G)}$ (with
$i=1,2,\ldots,c$) such that

(5.2)
$G=\overline{\gamma_{1}(G)}\geq\overline{\gamma_{2}(G)}=\overline{[G,G]}\geq\overline{\gamma_{3}(G)}=\overline{[[G,G],G]}\geq\ldots\geq\overline{\gamma_{c}(G)}\geq\overline{\gamma_{c+1}(G)}=1$

and $\overline{\gamma_{i}(G)}/\overline{\gamma_{i+1}(G)}$ are locally compact
abelian $p$-groups for all $i$. Note also that closed subgroups and quotients
of a periodic locally compact $p$-group of finite $p$-rank have finite
$p$-rank. This means that if $G$ has finite $p$-rank, then
$\overline{\gamma_{i}(G)}/\overline{\gamma_{i+1}(G)}$ are of the form of those
in Theorem 3.5, and in particular continuous automorphisms of
$\overline{\gamma_{i}(G)}/\overline{\gamma_{i+1}(G)}$ have finite topological
entropy. Now we do induction on $c$. Assume $c=1$. Then $G$ is a locally
compact abelian group of finite $p$-rank and the result is true by Theorem
3.5, because in this situation the continuous automorphisms of $G$ should have
finite topological entropy. Assume $c>1$ and that the result is true for all
periodic nilpotent locally compact $p$-groups of derived length at most $c-1$.
Then the continuous automorphisms of $N=\overline{\gamma_{c}(G)}$ have finite
topological entropy, since $N$ is abelian, but also those of $G/N$ have finite
topological entropy, since $G/N$ is a locally compact abelian $p$-group of
finite $p$-rank. From Addition Theorem for continuous automorphisms of totally
disconnected locally compact abelian groups (see [18, Addition Theorem 10], or
Corollary 3.3 (b)) we conclude that $AT(G,\varphi,N)$ holds for every
continuous automorphism $\varphi$ of $G$. The result follows. ∎

###### Proof of Theorem 1.5.

From Lemma 4.4, the Heisenberg group $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is a
periodic locally compact nonabelian $p$-group of nilpotency class two and
$\mathrm{rank}_{p}(\mathbb{H}_{n}(\mathbb{Q}_{p}))=2n$. Then we shall only
prove that $\mathbb{H}_{n}(\mathbb{Q}_{p})$ belongs to
$\mathfrak{E}_{<\infty}$, but not to $\mathfrak{E}_{0}$.

Assume that $n=1$. From [8, Theorem 6.8] we know that
$\mathbb{H}(\mathbb{Q}_{p})$ belongs to $\mathfrak{E}_{<\infty}$, but not to
$\mathfrak{E}_{0}$. Then there exists a subgroup $S$ of
$\mathbb{H}_{n}(\mathbb{Q}_{p})$ which is isomorphic to
$\mathbb{H}(\mathbb{Q}_{p})$ as topological group, for instance $S$ can be
realized putting in (4.1) the condition $a_{i}=b_{i}=0$ for all
$i=2,3,\ldots,n$. This is sufficient to show that
$\mathbb{H}_{n}(\mathbb{Q}_{p})$ cannot be in $\mathfrak{E}_{0}$, since it
contains a subgroup $S$ which is not in $\mathfrak{E}_{0}$. It remains to
check that $\mathbb{H}_{n}(\mathbb{Q}_{p})$ belongs to
$\mathfrak{E}_{<\infty}$ and we adapt the argument of [8, Proof of Theorem
6.8] for this scope.

Consider $\varphi\in\mathrm{End}(\mathbb{H}_{n}(\mathbb{Q}_{p}))$ and
$N=\ker\varphi$; we claim that $\mathsf{h}_{\mathsf{top}}(\varphi)<\infty$.

Assume that $N=1$. We claim that
$\varphi\in\mathrm{Aut}(\mathbb{H}_{n}(\mathbb{Q}_{p}))$. Since
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$ is fully invariant,
$\varphi_{|Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))}$ is injective, hence
$\varphi_{|Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))}$ is a continuous automorphism of
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$. In particular,
$\varphi^{-1}(Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))=Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$
and so
$\bar{\varphi}_{|\mathbb{H}_{n}(\mathbb{Q}_{p})/Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))}$
on $\mathbb{H}_{n}(\mathbb{Q}_{p})/Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$ is
injective. In fact it is a continuous automorphism of
$\mathbb{H}_{n}(\mathbb{Q}_{p})/Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))\simeq\mathbb{Q}^{2n}_{p}$.
Now $\mathbb{H}_{n}(\mathbb{Q}_{p})$ is a totally disconnected locally compact
group, which can be also realized as union of countably many compact sets, and
so $\varphi$ is a continuous automorphism by the Open Mapping Theorem [13,
Appendix 1, Exercise EA1.21]. We may apply Addition Theorems on closed normal
subgroups for continuous automorphisms of locally compact groups as per
Corollary 3.3 (c), concluding $\mathsf{h}_{\mathsf{top}}(\varphi)<\infty$ from
the fact that both
$\mathsf{h}_{\mathsf{top}}(\varphi_{|Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))})<\infty$
and
$\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{\mathbb{H}_{n}(\mathbb{Q}_{p})/Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))})<\infty$
by Theorem 3.5.

Now assume that $N=\ker\varphi\neq 1$. First we show that $N\cap
Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$ is nontrivial and then that
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))\subseteq N$. If there exists some $y\in
N\setminus Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$, then there exists
$x\in\mathbb{H}_{n}(\mathbb{Q}_{p})$ such that $[x,y]$ is nontrivial. This
implies that
$N\cap[\mathbb{H}_{n}(\mathbb{Q}_{p}),\mathbb{H}_{n}(\mathbb{Q}_{p})]$ is
nontrivial, because $[x,y]\in N$. The claim follows and $N\cap
Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$ is a nontrivial closed subgroup of
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))$, hence
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))/(N\cap Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))$
is torsion because nontrivial quotient of $\mathbb{Q}_{p}$. On the other hand,
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))/(N\cap
Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))\cong\varphi(Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))$
is a subgroup of $\mathbb{H}_{n}(\mathbb{Q}_{p})$ (up to continuous
isomorphisms), hence torsion-free. Consequently
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))/(N\cap Z(\mathbb{H}_{n}(\mathbb{Q}_{p})))$
is trivial, and the other claim $Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))\subseteq N$
follows. Since $N$ contains
$Z(\mathbb{H}_{n}(\mathbb{Q}_{p}))=[\mathbb{H}_{n}(\mathbb{Q}_{p}),\mathbb{H}_{n}(\mathbb{Q}_{p})]$,
we may apply Addition Theorems as per Corollary 3.3 (b), hence
$\mathsf{h}_{\mathsf{top}}(\varphi)=\mathsf{h}_{\mathsf{top}}(\bar{\varphi}_{\mathbb{H}_{n}(\mathbb{Q}_{p})/N})$
is finite by Theorem 3.5. Therefore the result follows.

∎

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