Introduction and statement of results
We consider automorphisms
for some identity neighbourhood
of expansive automorphisms
Our goal is to improve the understanding in the case
of non-compact groups. Special cases of expansive automorphisms are
Note that every automorphism α of a discrete group G
Our first main result generalizes [31, Proposition 6.1]
(devoted to the case of compact groups).
With a view towards our next result, recall that a topological group G is called topologically perfect
if its commutator group
Composition series play a central role in the study of contractive automorphisms . In the case of expansive automorphisms, composition series need not exist. However, certain substitutes are available.
of G such that
In addition, one may assume that all abelian, non-discrete
According to ,
a compact open subgroup
, where and ,
- (T2)The α-stable subgroups
, are closed in G.
here. The index
is called the scale of α; it is independent of the choice of the tidy subgroup V
is a subgroup of G called the associated contraction
group. In general,
of all torsion elements in the group
of divisible elements
are α-stable closed subgroups
internally as a topological group,
if we endow
Recall that the closure
Let G be a totally disconnected, locally compact group
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
Let G be a p-adic Lie group
which is linear in the sense that there
exists an injective continuous homomorphism
In many examples,
1 Preliminaries and basic facts
for any compact subset
We shall also need the so-called Levi factor
it is known that
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,
and for each compact open subgroup such that ,
- (c)G has a compact open subgroup V such that
is open in G,
is expansive, for each α -stable subgroup .
(a) Since α is expansive, there exists
an identity neighbourhood V such that
(b) By [1, Proposition 3.16],
(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
Let α be an automorphism of a totally disconnected, locally compact group G. The following facts are useful:
- (a)The closure of
in G is (see [1, Corollary 3.30] if G is metrizable; the general case follows with ).
(see [1, Corollary 3.27] if G is metrizable; the general case follows with ).
and are dense in (see [31, Theorem 4.1 (v) and Proposition 5.4 (i)]).
is a closed α-stable normal subgroup, the canonical quotient morphism and the automorphism of induced by α, then (see [1, Theorem 3.8] if G is metrizable,  in the general case).
is closed if and only if is closed, if and only if 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 ).
- (f)The so-called parabolic subgroupnormalizes
(see [1, Proposition 3.4]). Hence also its subgroups and normalize .
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:
is expansive if and only if its restriction
to the Levi factor is expansive.
is closed, then α is expansive if and only if is open in G, if and only if is discrete.
(a) In view of Lemma 1.1 (e), we only need to show that
(b) If α is expansive, then
If α is an automorphism of a totally disconnected,
compact group G, then
Recall that a group G is called a torsion group of finite
exponent if there exists
Let α be an expansive automorphism of a totally disconnected, compact group G. Then the following holds:
is open in G (see [31, Lemma 5.1]), whence also and are open in G (by 1.2 (b)).
- (b)G is a torsion group of finite exponent. [By (a) and
is a normal subgroup and is finite, hence a torsion group of finite exponent. It therefore suffices to show that is a torsion group of finite exponent. This is immediate from [31, Proposition 4.4, Theorem 6.2, Proposition 6.3]).
and are normal in G, the set is an open α-stable subgroup of G contained in . Thus by [31, Corollary 4.3].
Let α be an automorphism of a totally disconnected,
locally compact group G.
- (a)Then nub of
is contained in .
, then .
(a) Using 1.2 (b) twice, we deduce that
(using 1.2 (c)).
We shall also need certain facts concerning contractive automorphisms.
Let α be a contractive automorphism of a topological group G.
, then G is infinite and non-discrete. [If , then we have for all n and , entailing that the topological group 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 there exists such that for all (see [24, Lemma 1.4 (iv)]). Moreover, G is non-compact (unless ); see [24, Section 3.1].
Let α be a contractive automorphism of a totally disconnected, locally compact group G. Then the following holds:
, then is an integer (see [10, Proposition 1.1 (e)]).
is a series of α-stable closed subgroups of G such that is a proper normal subgroup of for all , then n is bounded by the number of prime factors of (see [10, Lemma 3.5]).
as a topological group for a divisible torsion free group and a torsion group of finite exponent (cf. [10, Theorem B]).
is an α-stable closed normal subgroup, the canonical quotient morphism and the automorphism of induced by α, then[The inclusions and are clear. Since we have and , equality follows.]
If an automorphism
2 Making contraction groups locally compact
The problem of refining group topologies on contraction groups was studied by Siebert . 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
If τ is totally disconnected,
is continuous. Reversing the roles of
See also [25, Proposition 9] for the following fact.
Let G be a totally disconnected, locally compact
group. If an automorphism
The right-shift α is an automorphism
of the compact group
Let G be a topological group and
can be made locally compact for each closed α -stable subgroup H of G (or ), and carries the topology induced by .
is a continuous homomorphism to a topological group H admitting an automorphism such that , then is contractive, can be made locally compact andis a topological quotient map.
(a) Let τ and
(b) Let σ be the topology on
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
We shall use a simple fact.
Let K be a compact group and
Let α be an automorphism of
a totally disconnected, locally compact
group G. If
(a) By Lemma 2.6 (a),
(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
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
After passing to a subsequence, we may assume that
Conversely, assume that both
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
is open in G, in which case is topologically perfect.
Note that C is an open subgroup of G as it
contains the open set
of α-stable closed subgroups of G, let
over all series Σ
For an index
thus we have that
(by Lemma 4.1, only these two cases
is a series of α-stable closed subnormal
subgroups such that all non-discrete subfactors
are abelian or topologically perfect.
Using the recent theory of elementary groups ,
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,
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
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
Case 2: If
5 Proof of Theorem C
We prove Theorem C and record some related results.
Let G be a totally disconnected,
locally compact group and α an automorphism of G
The nub of an expansive automorphism
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
let I be the set of all indices
for some , together with a contractive linear automorphism not admitting non-trivial proper β -stable vector subspaces,
for some , together with for a contractive linear automorphism not admitting non-trivial proper β -stable vector subspaces,
with the right-shift,
with the left-shift,
with the right-shift.
Throughout the rest of the proof,
Let G be a totally disconnected, locally compact group
is a continuous, open homomorphism with discrete
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
It can happen that
Given a non-trivial finite abelian group
is a continuous surjective homomorphism.
Restricted to the compact open subgroup
Another property can be observed.
Let G be a totally disconnected,
locally compact group that is abelian, and
we see that
7 Example: p-adic Lie groups
(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.
If α is expansive,
is an analytic diffeomorphism.
Using [24, Lemma 3.2 (i)],
we find compact open subgroups
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
, there is a unique continuous homomorphism such that . Moreover, .
is an -stable Lie subalgebra, then there exists an α -stable Lie subgroup H of G with .
(b) We may work with the isomorphic group
Let G be a linear p-adic Lie group.
Assume that G is generated by
After replacing G with an open subgroup, we may assume
that G is generated by
Now assume that
To prove the claim, let
If G is a totally disconnected, locally compact group which is a
nilpotent group, let
Note that we can easily achieve that
The following example shows that even for nilpotent
p-adic Lie groups with an expansive automorphism α,
Let G be a closed subgroup of
Replacing G by the group generated by
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.
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
Let β be the inner automorphism of G given by
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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Thus C is the subgroup
Let Θ be the set
of continuous homomorphisms from