## Introduction and statement of results

We consider automorphisms
*G* which are
*expansive* in the sense that

for some identity neighbourhood *compact*
groups were studied in [14, 22] and
recently
in [31].
The importance of expansive automorphisms
for the theory of general automorphisms is highlighted by the fact
that every automorphism α of a totally disconnected compact group *G*
is a projective limit

of expansive automorphisms *G* such that

Our goal is to improve the understanding in the case
of non-compact groups. Special cases of expansive automorphisms are
automorphisms *contractive* in
the sense that

Note that every automorphism α of a discrete group *G*
(e.g.,

Our first main result generalizes [31, Proposition 6.1]
(devoted to the case of compact groups).
As usual,
a subset *α-stable* if

*Let *

With a view towards our next result, recall that a topological group *G* is called *topologically perfect*
if its commutator group *G*.

Composition series play a central role in the study of contractive automorphisms [10]. In the case of expansive automorphisms, composition series need not exist. However, certain substitutes are available.

*If *

*of G such that *

In addition, one may assume that all abelian, non-discrete
factors

According to [30],
a compact open subgroup *tidy for α*
if it has the following properties:

- (T1)
, where$V={V}_{+}{V}_{-}$ and${V}_{+}:={\bigcap}_{n=0}^{\infty}{\alpha}^{n}\left(V\right)$ ,${V}_{-}:={\bigcap}_{n=0}^{\infty}{\alpha}^{-n}\left(V\right)$ - (T2)The α-stable subgroups
,${V}_{++}:={\bigcup}_{n\in {\mathbb{N}}_{0}}{\alpha}^{n}\left({V}_{+}\right)$ are closed in${V}_{--}:={\bigcup}_{n\in {\mathbb{N}}_{0}}{\alpha}^{-n}\left({V}_{-}\right)$ *G*.

Note that

and

here. The index

is called the *scale of α*; it is independent of the choice of the tidy subgroup *V*
(see [30]).
Following [31],
the intersection *V* which are tidy for α
is called the *nub* of α;
it is a compact, α-stable subgroup of *G*.

If *G*,
then

is a subgroup of *G* called the associated *contraction
group*. In general, *can be made locally compact*,
i.e., it can be
refined to a totally disconnected,
locally compact group topology

of all torsion elements in the group

of divisible elements
are α-stable closed subgroups
of

internally as a topological group,
if we endow

Recall that the closure
*G*
plays a role in the structure
theory of totally disconnected,
locally compact groups; for example, the scale

*Let G be a totally disconnected, locally compact group
and let *

We mention that
the nub of an expansive automorphism α
need not have an open normalizer in *G*
(see Remark 5.2), in which case
not both of

Classes of examples are also considered.
An analytic automorphism α of a Lie group *G* over a totally disconnected
local field *G* is then nilpotent (see Proposition 7.1).
In the case of *p*-adic Lie groups
for a prime number *p*,
we obtain:

*Let G be a p-adic Lie group
which is linear in the sense that there
exists an injective continuous homomorphism *

If *G*, then *G* (see
Proposition 1.1).
This is essential for our studies;
for instance, it allows finiteness properties of
locally compact contraction groups (viz. bounds for the length
of series of α-stable closed subgroups)
to be exploited in the proof of Theorem B.

In many examples, *G* (for instance,
for all expansive automorphisms of closed subgroups

with primes

*Let $\alpha :{G}_{p,q}\to {G}_{p,q}$ be the conjugation by t.
Then α is expansive but ${U}_{\alpha}{U}_{{\alpha}^{-1}}$ is not a
subgroup of ${G}_{p,q}$.*

## 1 Preliminaries and basic facts

We write *J* is a finite set, we let *X* is a proper subset of *Y*. As usual, we write *N* is a normal subgroup of *G*. All topological groups
considered in this article are assumed Hausdorff, and locally
compact topological groups are simply called locally compact groups.
Totally disconnected, locally compact non-discrete topological fields
(like the field of *p*-adic numbers) will be called local fields
(see [27] for further information). See [3] and
[23] for basic information on Lie groups over local (and more
general complete ultrametric) fields (which we always assume
finite-dimensional). If we say that α is an automorphism of a
topological group, then we assume that both α and
*G*. If a subgroup *N* is called *topologically characteristic*. A
topological group *G* is called *topologically perfect* if its
commutator group *G*. If *F* is a finite group
and *X* a set, we write *quotient morphisms*.
A topological space *X* is called *σ-compact*
if it is a union

If *G* and denote
the module of α by

for any compact subset

We shall also need the so-called *Levi factor*

it is known that *G* (see [1, p. 224]).
The following lemma compiles basic facts
concerning expansive automorphisms.

*If α is an expansive automorphism
of a totally disconnected,
locally compact group G,
then the following holds:*

- (a)
*G**is metrizable and has an*α*-stable,*σ*-compact open subgroup,* - (b)
${V}_{--}={U}_{\alpha}$ *and*${V}_{++}={U}_{{\alpha}^{-1}}$ *for each compact open subgroup*$V\subseteq G$ *such that* ,${V}_{0}=\left\{1\right\}$ - (c)
*G**has a compact open subgroup**V**such that*$V={V}_{+}{V}_{-}$ *and* ,${V}_{0}=\left\{1\right\}$ - (d)
${U}_{\alpha}{U}_{{\alpha}^{-1}}$ *is open in**G*, - (e)
${\alpha |}_{H}$ *is expansive, for each*α*-stable subgroup* .$H\subseteq G$

(a) Since α is expansive, there exists
an identity neighbourhood *V* such that
*V* with a smaller compact identity neighbourhood if necessary,
we may assume that *V* is compact.
Since *V* is compact and
*V*
with intersection *G* is metrizable.
The subgroup of *G* generated by

(b) By [1, Proposition 3.16],
we have

(c) and (d) Using expansiveness and van Dantzig’s Theorem
[11, Theorem 7.7],
we find a compact open subgroup

By [29, Lemma 1], there
exists *gh*. Hence *gh* in its interior and thus

(e) If *H*
and

The first statement of Lemma 1.1 (a) also follows from [15, Lemma 2.4].

Let α be an automorphism of a totally disconnected,
locally compact group *G*.
The following facts are useful:

- (a)The closure of
in${U}_{\alpha}$ *G*is (see [1, Corollary 3.30] if$\overline{{U}_{\alpha}}={U}_{\alpha}nub\left(\alpha \right)$ *G*is metrizable; the general case follows with [13]). - (b)
(see [1, Corollary 3.27] if$nub\left(\alpha \right)=\overline{{U}_{\alpha}}\cap \overline{{U}_{{\alpha}^{-1}}}$ *G*is metrizable; the general case follows with [13]). - (c)Both
and$nub\left(\alpha \right)\cap {U}_{\alpha}$ are dense in$nub\left(\alpha \right)\cap {U}_{{\alpha}^{-1}}$ (see [31, Theorem 4.1 (v) and Proposition 5.4 (i)]).$nub\left(\alpha \right)$ - (d)If
is a closed α-stable normal subgroup,$N\subseteq G$ the canonical quotient morphism and$q:G\to G/N$ the automorphism of$\overline{\alpha}$ induced by α, then$G/N$ (see [1, Theorem 3.8] if$q\left({U}_{\alpha}\right)={U}_{\overline{\alpha}}$ *G*is metrizable, [13] in the general case). - (e)
is closed if and only if${U}_{\alpha}$ is closed, if and only if${U}_{{\alpha}^{-1}}$ *G*has small subgroups tidy for α, i.e., every identity neighbourhood of*G*contains some tidy subgroup (see [1, Theorem 3.32] if*G*is metrizable; the general case can be deduced using the techniques from [13]). - (f)The so-called
*parabolic subgroup*normalizes${P}_{\alpha}:=\{g\in G:\{{\alpha}^{n}\left(g\right):n\in {\mathbb{N}}_{0}\}\text{is relatively compact}\}$ (see [1, Proposition 3.4]). Hence also its subgroups${U}_{\alpha}$ and${M}_{\alpha}={P}_{\alpha}\cap {P}_{{\alpha}^{-1}}$ normalize$nub\left(\alpha \right)\subseteq {M}_{\alpha}$ .${U}_{\alpha}$

The following results help to show that certain automorphisms are expansive.

*Let α be an automorphism of a totally disconnected, locally
compact group G. Then the following holds:*

- (a)
α
*is expansive if and only if its restriction*${\alpha |}_{{M}_{\alpha}}$ *to the Levi factor is expansive.* - (b)
*If*${U}_{\alpha}$ *is closed, then*α*is expansive if and only if*${U}_{\alpha}{U}_{{\alpha}^{-1}}$ *is open in**G**, if and only if*${M}_{\alpha}$ *is discrete.*

(a) In view of Lemma 1.1 (e), we only need to show that
if *I* is
α-stable and

Thus

(b) If α is expansive, then *G* and the product map

is a homeomorphism (by 1.2 (e)
above and part (f)
from the theorem in [5]).
Hence

If α is an automorphism of a totally disconnected,
*compact* group *G*, then
*G* (since

Recall that a group *G* is called a torsion group *of finite
exponent* if there exists *H*, then
also *g* in the closure *G* in *H*, and
thus also

Let α be an expansive automorphism of a totally disconnected,
compact group *G*. Then the following holds:

- (a)
is open in$nub\left(\alpha \right)$ *G*(see [31, Lemma 5.1]), whence also and$\overline{{U}_{\alpha}}$ are open in$\overline{{U}_{{\alpha}^{-1}}}$ *G*(by 1.2 (b)). - (b)
*G*is a torsion group of finite exponent. [By (a) and 1.4, is a normal subgroup and$nub\left(\alpha \right)$ is finite, hence a torsion group of finite exponent. It therefore suffices to show that$G/nub\left(\alpha \right)$ is a torsion group of finite exponent. This is immediate from [31, Proposition 4.4, Theorem 6.2, Proposition 6.3]).$nub\left(\alpha \right)$ - (c)Since
and${U}_{\alpha}$ are normal in${U}_{{\alpha}^{-1}}$ *G*, the set is an open α-stable subgroup of${U}_{\alpha}{U}_{{\alpha}^{-1}}$ *G*contained in . Thus$nub\left(\alpha \right)$ by [31, Corollary 4.3].$nub\left(\alpha \right)={U}_{\alpha}{U}_{{\alpha}^{-1}}$

*Let α be an automorphism of a totally disconnected,
locally compact group G.
Let *

- (a)
*Then nub of*${\alpha |}_{H}$ *is contained in* .$nub\left(\alpha \right)$ - (b)
*If*$nub\left(\alpha \right)\subseteq H$ *, then* .$nub\left({\alpha |}_{H}\right)=nub\left(\alpha \right)$

(a) Using 1.2 (b) twice, we deduce that

(b) Since

(using 1.2 (c)).
Likewise,

Since

We shall also need certain facts concerning contractive automorphisms.

Let α be a contractive automorphism of a topological
group *G*.

- (a)If
, then$G\ne \left\{1\right\}$ *G*is infinite and non-discrete. [If , then we have$x\in G\setminus \left\{1\right\}$ for all${\alpha}^{n}\left(x\right)\ne 1$ *n*and , entailing that the topological group${\alpha}^{n}\left(x\right)\to 1$ *G*is not discrete and hence infinite.] - (b)If
*G*is locally compact, then α is*compactly contractive*, i.e., for each identity neighbourhood and compact set$U\subseteq G$ there exists$K\subseteq G$ such that$m\in \mathbb{N}$ for all${\alpha}^{n}\left(K\right)\subseteq U$ (see [24, Lemma 1.4 (iv)]). Moreover,$n\ge m$ *G*is non-compact (unless ); see [24, Section 3.1].$G=\left\{1\right\}$

Let α be a contractive automorphism
of a totally disconnected,
locally compact group *G*. Then the following holds:

- (a)If
, then$G\ne \left\{1\right\}$ is an integer${\Delta}_{G}\left({\alpha}^{-1}\right)$ (see [10, Proposition 1.1 (e)]).$\ge 2$ - (b)If
is a series of α-stable closed subgroups of$G={G}_{0}\supset {G}_{1}\supset {G}_{2}\supset \cdots \supset {G}_{n}=\left\{1\right\}$ *G*such that is a proper normal subgroup of${G}_{j}$ for all${G}_{j-1}$ , then$j\in \{1,\dots ,n\}$ *n*is bounded by the number of prime factors of (see [10, Lemma 3.5]).${\Delta}_{G}\left({\alpha}^{-1}\right)$ - (c)
as a topological group for a divisible torsion free group$G=div\left(G\right)\times tor\left(G\right)$ and a torsion group$div\left(G\right)$ of finite exponent (cf. [10, Theorem B]).$tor\left(G\right)$ - (d)If
is an α-stable closed normal subgroup,$N\subseteq G$ the canonical quotient morphism and$q:G\to G/N$ the automorphism of$\overline{\alpha}$ induced by α, then$G/N$ [The inclusions$q\left(div\left(G\right)\right)=div\left(G/N\right)\hspace{1em}\text{and}\hspace{1em}q\left(tor\left(G\right)\right)=tor\left(G/N\right).$ and$q\left(div\left(G\right)\right)\subseteq div\left(G/N\right)$ are clear. Since we have$q\left(tor\left(G\right)\right)\subseteq tor\left(G/N\right)$ and$G/N=q\left(div\left(G\right)tor\left(G\right)\right)=q\left(div\left(G\right)\right)q\left(tor\left(G\right)\right)$ , equality follows.]$G/N=div\left(G/N\right)\times tor\left(G/N\right)$

If an automorphism

## 2 Making contraction groups locally compact

The problem of refining group topologies on contraction groups was studied by Siebert [25]. The following special case is useful for our purposes.

Let *G* be a topological group, with topology τ, and let

be a contractive automorphism.
We say that *G*) *can be made locally compact*
if there exists a locally compact topology *G* making it a topological group
such that *G*, endowed with the topology

If τ is totally disconnected,
then also

The topology

*The topology *

Assume that *G*
with the same properties as

is continuous. Reversing the roles of *V* with the closure
of its interior *V*.
Then *V* is also compact in *W* is also compact in *V* is the countable union of the closed subsets
*V*.
Since *V*, we deduce
that *W* has non-empty interior *U* is an identity neighbourhood in

See also [25, Proposition 9] for the following fact.

*Let G be a totally disconnected, locally compact
group. If an automorphism *

Let

The group
*G*
over a local field
(see Proposition 13.3 (b)
in the extended preprint version of [7]).

The right-shift α is an automorphism
of the compact group *p*-adic integers.
The contraction group

*Let G be a topological group and *

- (a)
$(H,{\alpha |}_{H})$ *can be made locally compact for each closed*α*-stable subgroup**H**of**G**(or*${G}^{*}$ *), and*${H}^{*}$ *carries the topology induced by* .${G}^{*}$ - (b)
*If*$\varphi :G\to H$ *is a continuous homomorphism to a topological group**H**admitting an automorphism*$\beta :H\to H$ *such that*$\beta \circ \varphi =\varphi \circ \alpha $ *, then*${\beta |}_{\varphi \left(G\right)}$ *is contractive,*$\varphi \left(G\right)$ *can be made locally compact and*${G}^{*}\to \varphi {\left(G\right)}^{*},x\mapsto \varphi \left(x\right)$ *is a topological quotient map.*

(a) Let τ and *H* is closed in *H* is a locally compact group
in the topology σ
on *H* induced by *G* and turns

(b) Let σ be the topology on *H*. Moreover,

is continuous, the map

The following observation is crucial for many of our arguments.

*Let α be an expansive automorphism
of a totally disconnected, locally compact group G
and let*

*be α-stable closed subgroups of G such that
*

*where *

Let

(resp.,

Since α is expansive, it follows that

We shall use a simple fact.

*Let K be a compact group and *

For each *D* by hypothesis, we see that

If *F*
separate points on

*Let α be an automorphism of
a totally disconnected, locally compact
group G. If *

- (a)
${U}_{\alpha}\cap nub\left(\alpha \right)={T}_{\alpha}\cap nub\left(\alpha \right)$ *;* - (b)
$nub\left(\alpha \right)=\overline{{T}_{\alpha}\cap nub\left(\alpha \right)}$ *;* - (c)
$\overline{{T}_{\alpha}}\cap {U}_{\alpha}={T}_{\alpha}$ *;* - (d)
.$\overline{{T}_{\alpha}}={T}_{\alpha}nub\left(\alpha \right)$

*If both *

- (e)
.$nub\left(\alpha \right)=\overline{{T}_{\alpha}}\cap \overline{{T}_{{\alpha}^{-1}}}$

(a) By Lemma 2.6 (a),

and hence

(b) As

(c) The subgroup

(d) We have

(using (c) for the last equality).

(e) We have

(see 1.2 (b)) and

(using (b)). ∎

*Let α be an automorphism of a locally compact group G and let
*

Let *S* of *S* is an α-stable open subgroup of *S* is closed). Hence

## 3 Proof of Theorem A

We now prove Theorem A.

Let *G* be a totally disconnected, locally compact group, α
an expansive automorphism of *G* and *G* with *U* be a subgroup of *U* is an *G* with

Set *G*, with

*There exists *
Indeed, if this were false, we could find
a subsequence

After passing to a subsequence, we may assume that *N* is open in *n*. Since

Conversely, assume that both ^{1}
Then there is an open identity neighbourhood *Q*, we may assume that *V* with

entailing
that *N*.
Since

we deduce that

## 4 Proof of Theorem B

The following lemma is useful.

*Let α be an expansive automorphism
of a totally disconnected, locally compact group G
such that every α-stable, closed normal
subgroup *

- (a)
$\overline{[C,C]}$ *is open in**G**, in which case*$C=\overline{[C,C]}$ *is topologically perfect.* - (b)
$\overline{[C,C]}$ *is discrete.*

Note that *C* is an open subgroup of *G* as it
contains the open set *C*, whence it is topologically characteristic
in *G* and hence α-stable and normal.
Therefore *C*
is the smallest topologically characteristic subgroup of *G*
which contains

Define

of α-stable closed subgroups of *G*, let

over all series Σ
exists. Let *N* will be open in *N*.

For an index

thus we have that

(by Lemma 4.1, only these two cases
can occur).
So

is a series of α-stable closed subnormal
subgroups such that all non-discrete subfactors
are abelian or topologically perfect.
Since

Using the recent theory of elementary groups [28],
slightly more detailed information on the factor groups can be obtained,
in the case of second countable groups.
Recall that the class of
elementary groups is the smallest class of totally disconnected,
second
countable, locally compact groups that contains all countable discrete groups and all second countable pro-finite groups,
and is closed under extensions as well as countable increasing unions. A totally disconnected, second countable, locally compact group *G* is called *elementary-free*
if all of its elementary closed normal subgroups and all of its
elementary Hausdorff quotient groups are trivial [28, Definition 7.14].
If α is an expansive automorphism of a totally disconnected, locally compact non-trivial group *G*
and *G* does not have closed α-stable subgroups
except for *G* and *simple expansion group*.
Note that if *G* is a non-trivial elementary-free group and α an expansive automorphism
of *G* such that every α-stable closed normal subgroup of *G* is discrete
or open, then

If *G* is second countable in Theorem B, then one can achieve there that each of the quotient groups

In fact, let us consider a topologically perfect factor *Q* such that

Case 1:
If

Case 2: If

## 5 Proof of Theorem C

We prove Theorem C and record some related results.

Since *G* with *G*.
Since *G*.
Therefore also *G*.
Since

Hence *G* is locally compact and the product
map *G*.∎

*Let G be a totally disconnected,
locally compact group and α an automorphism of G
such that *

Since

The nub of an expansive automorphism
*G*. To
see this, let *F* be a finite group which is a semidirect product
*N* and a subgroup *H* which is
not normal in *F* (e.g., *F* might be the dihedral group *G* be the group of all *G* with the topology
making it the direct product topological space of the restricted
product *M* and the compact group *G* is a topological
group, being the ascending union of the open subgroups
*G*.
We have

and

Thus *H* is not normal in *F*,
we see that *G*.
If the normalizer *G*,
then (being α-stable), it would contain
the dense subgroup *G* and hence coincide with *G*
(a contradiction). Thus

## 6 Abelian expansion groups

We show that, after passing to a refinement if necessary, only abelian, non-discrete groups of a special form will occur in Theorem B.

In the situation of
Theorem B,
let *I* be the set of all indices

Let

*Let *

*A*is discrete. Then there exists a prime number

*p*such that

- (a)
${\mathbb{Q}}_{p}^{n}$ *for some*$n\in \mathbb{N}$ *, together with a contractive linear automorphism*$\beta :{\mathbb{Q}}_{p}^{n}\to {\mathbb{Q}}_{p}^{n}$ *not admitting non-trivial proper*β*-stable vector subspaces,* - (b)
${\mathbb{Q}}_{p}^{n}$ *for some*$n\in \mathbb{N}$ *, together with*${\beta}^{-1}$ *for a contractive linear automorphism*$\beta :{\mathbb{Q}}_{p}^{n}\to {\mathbb{Q}}_{p}^{n}$ *not admitting non-trivial proper*β*-stable vector subspaces,* - (c)
${C}_{p}^{\left(-\mathbb{N}\right)}\times {C}_{p}^{{\mathbb{N}}_{0}}$ *with the right-shift,* - (d)
${C}_{p}^{\left(-\mathbb{N}\right)}\times {C}_{p}^{{\mathbb{N}}_{0}}$ *with the left-shift,* - (e)
${C}_{p}^{\mathbb{Z}}$ *with the right-shift.*

Let *A* is of the form described in (b)
whenever

Throughout the rest of the proof,
assume that

Since *A*, it is either all
of *A* or discrete. Being also compact,
it is finite in the latter case,
and thus

Case *A*
(using 1.2 (a)).
If *A* is of the
form described in (d)
if

Case *A* is compact and is irreducible in the sense of
[31, Definition 6.1] as all its proper α-stable closed
(normal) subgroups are finite, and moreover *A* is infinite (as
*F*. Since *A* is
abelian, *p* and thus *A* is of the form
described in (e).
∎

Let *G* be a totally disconnected, locally compact group
and let *G* is abelian, then the map

is a continuous, open homomorphism with discrete
kernel.
For non-abelian *G*, the map still has open
image (see Lemma 1.1 (d)),
is a local homeomorphism,
and equivariant with respect to the natural
left and right actions of

[To see this, let *G* and

It can happen that *G*,
but

Given a non-trivial finite abelian group

with

is a continuous surjective homomorphism.
Restricted to the compact open subgroup *q* is an isomorphism of topological groups.
Hence *q* is open, has discrete kernel,
and is a quotient morphism.
Finally,

Another property can be observed.

*Let G be a totally disconnected,
locally compact group that is abelian, and *

Since *G*, we
need only show that *G* with its α-stable subgroup *V*, we may
therefore assume that

we see that

internally as an abstract group.
By equation (6.1),
the torsion subgroup of *G* is

## 7 Example: *p*-adic Lie groups

Let *E* is a finite-dimensional

Then

with

(cf. [8, Lemma 2.5]).

If *G* is a Lie group over

*Let α be an analytic automorphism
of a Lie group G over a local field.
Then the following conditions are equivalent:*

- (a)
α
*is expansive.* - (b)
$\beta :=L\left(\alpha \right):L\left(G\right)\to L\left(G\right)$ *is expansive.* - (c)
.$1\notin R\left(\beta \right)$

*If α is expansive,
then *

(a) *G* contains a so-called centre manifold *W*
around the fixed point 1 of α,
which can be chosen as a
submanifold of *G* contained in *V*
that is stable under α
and satisfies

(b)

(c) *G* with Lie algebras *VW* is open in *G* and
the restriction

is an analytic diffeomorphism.
Using [24, Lemma 3.2 (i)],
we find compact open subgroups *PQ* is an open identity neighbourhood
in *G*
and we now show that

*Final assertion.*
If (c) holds, then β is a Lie algebra automorphism of

For each continuous homomorphism

*Let α be a contractive automorphism
of a p-adic Lie group G.
Then the following holds:*

- (a)
*For each*$g\in G$ *, there is a**unique continuous homomorphism*${\theta}_{g}:{\mathbb{Q}}_{p}\to G$ *such that*${\theta}_{g}\left(1\right)=g$ *. Moreover,* .$\{{\theta}_{g}^{\prime}\left(0\right):g\in G\}=L\left(G\right)$ - (b)
*If*$\U0001d525\subseteq L\left(G\right)$ *is an*$L\left(\alpha \right)$ *-stable Lie subalgebra, then there exists an*α*-stable Lie subgroup**H**of**G**with* .$L\left(H\right)=\U0001d525$

(a) Let *ng* (in the vector space

(b) We may work with the isomorphic group
*G*.
Now

*Let G be a linear p-adic Lie group.
Assume that G is generated by *

Let *G*,
we see that

After replacing *G* with an open subgroup, we may assume
that *G* is generated by *G* is nilpotent in this case,
by induction on the dimension
*G* as a *p*-adic manifold.
If *G* is discrete, whence

Now assume that *G* with an isomorphic
group, we may assume that *G* is a subgroup of *H* is in
the centre *G*. If this is true, then
^{3}
By induction,
*G*.

To prove the claim, let

If *G* is a totally disconnected, locally compact group which is a
nilpotent group, let *G* and

*If *

*7.5*, then

*G*.

If

we have

Note that we can easily achieve that
*G* with
its open subgroup generated by

The following example shows that even for nilpotent
*p*-adic Lie groups with an expansive automorphism α,
the set *p*-adic Lie group that admits
expansive automorphisms but does not
admit any contractive automorphism. In fact,
the group has a closed discrete commutator group
which is characteristic and hence would inherit a contractive
automorphism (contradicting the fact that non-trivial contraction groups
are non-discrete).

Let *p*-adic Heisenberg group
whose binary operation is given by

for all *N* is a compact central
subgroup of *H*. Identify

for all *p*-adic Lie group
*G* with

*Let G be a closed subgroup of *

Replacing *G* by the group generated by *G* is a closed nilpotent
subgroup of *G*. Then

For *V*
be the maximal vector subspace of *V* is a closed
α-stable central subgroup of *G*. The automorphism *V* is the maximal vector
subspace of *p*-adic
vector space *p*-adic vector space *p*-adic analytic group is compact, compact *p*-adic
analytic groups do not admit expansive automorphisms unless finite,
hence *p*-adic algebraic) group, the result follows by induction.
∎

In the case of linear *p*-adic Lie groups, even if

Let *H* be the 3-dimensional *p*-adic Heisenberg group defined as
in Remark 7.7.
For

for

and

Thus *H*,
neither

## 8 Example: Baumslag–Solitar groups

Throughout this section, we fix primes

be the Baumslag–Solitar group. Then

By [4, Proposition 8.1], *K* contains an open subgroup *a* to

Let

for all *G*. Since

sending *G* which would also be
denoted by ϕ. Since

Let β be the inner automorphism of *G* given by

In case *G* is solvable.^{4}
Suppose *N* with ^{5}
Now the group generated by *t* and *N* is an open subgroup of *G*
containing both *t* and *a*, hence *t* normalizes *N*).
This implies that ϕ is an isomorphism.
∎

The second author wishes to thank DAAD and Institut für Mathematik, Universität Paderborn, Paderborn, Germany for providing the facilities during his stay. Comments by the referees led to improvements of the presentation and the inclusion of Remark 4.2.

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## Footnotes

^{1}

Compare also [31, Proposition 6.1].
The compactness of *G* assumed there is inessential
for this part of the proof of [31, Proposition 6.1].

^{2}

Thus *C* is the subgroup
generated by

^{3}

Let Θ be the set
of continuous homomorphisms from