Expansive automorphisms of totally disconnected, locally compact groups

Helge Glöckner 1  and C. R. E. Raja 2
  • 1 Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
  • 2 Stat-Math Unit, Indian Statistical Institute (ISI), 8th Mile Mysore Road, Bangalore 560 059, India
Helge Glöckner and C. R. E. Raja

Abstract

We study topological automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that

nαn(U)={1}

for some identity neighbourhood UG. Notably, we prove that the automorphism induced by an expansive automorphism α on a quotient group G/N modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups

Uα:={gG:αn(g)1 as n}.

If α is expansive, then UαUα-1 is an open identity neighbourhood in G. We give examples where UαUα-1 fails to be a subgroup. However, UαUα-1 is an α-stable, nilpotent open subgroup of G if G is a closed subgroup of GLn(p). Further results are devoted to the divisible and torsion parts of Uα, and to the so-called “nub” nub(α)=Uα¯Uα-1¯ of an expansive automorphism.

Introduction and statement of results

We consider automorphisms α:GG (thus α is a group automorphism such that both α and α-1 are continuous) of a totally disconnected locally compact topological group G which are expansive in the sense that

nαn(U)={1}

for some identity neighbourhood UG. Expansive automorphisms of totally disconnected, 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

α=limαj

of expansive automorphisms αj of certain Hausdorff quotient groups G/Nj of G such that G=limG/Nj (see [31]).

Our goal is to improve the understanding in the case of non-compact groups. Special cases of expansive automorphisms are automorphisms α:GG which are contractive in the sense that αn(g)1 as n for each gG (see Remark 1.10). The structure of totally disconnected, locally compact groups admitting contractive automorphisms was elucidated in [10] (building on earlier works like [24] and [26]), and the results obtained there can also be used as tools in the investigation of expansive automorphisms (as we shall see). Further tools come from the structure theory of totally disconnected, locally compact groups ([29, 30]), in which contractive automorphisms play an important role (as worked out in [1]).

Note that every automorphism α of a discrete group G (e.g., α=idG) is expansive (as we may choose U={1} then). Therefore discrete groups and their automorphisms are part of the theory of expansive automorphisms. As a consequence, groups permitting expansive automorphisms need not have any particular algebraic properties. However, there are topological implications: a totally disconnected, locally compact group admitting an expansive automorphism α is always metrizable (cf. [15]) and has a second countable, α-stable open subgroup (Lemma 1.1 (a)).

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

Let α:GG be an automorphism of a totally disconnected, locally compact group, let NG be an α-stable closed normal subgroup and let α¯:G/NG/N, gNα(g)N be the automorphism of G/N induced by α. Then α is expansive if and only if both α|N and α¯ are expansive.

With a view towards our next result, recall that a topological group G is called topologically perfect if its commutator group [G,G] is dense in 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 α:GG is an expansive automorphism of a totally disconnected, locally compact group G, then there exist α-stable closed subgroups

G=G0G1Gn={1}

of G such that Gj is normal in Gj-1 for j{1,,n} and every αj-stable closed normal subgroup of Gj-1/Gj is discrete or open, where αj denotes the induced automorphism Gj-1/GjGj-1/Gj, gGjα(g)Gj. Moreover, one can achieve this in such a way that each of the quotient groups Gj-1/Gj is discrete, abelian or topologically perfect.

In addition, one may assume that all abelian, non-discrete factors Gj-1/Gj are simple contraction groups with respect to the automorphism αj or its inverse, or isomorphic to an infinite power Cp of a cyclic group of prime order, endowed with the right-shift (cf. Remark 6.1 and Proposition 6.2). For second countable groups, more detailed information on the perfect factors is available (see Remark 4.2). The proofs of Theorem A and Theorem B hinge on the fact that there is a bound on the number of non-discrete factors in series for (G,α) (see Proposition 2.7).

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

  1. (T1)V=V+V-, where V+:=n=0αn(V) and V-:=n=0α-n(V),
  2. (T2)The α-stable subgroups V++:=n0αn(V+), V--:=n0α-n(V-) are closed in G.

Note that

V+α(V+)α2(V+)

and

V-α-1(V-)α-2(V-)

here. The index

s(α):=[α(V+):V+]

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

If α:GG is an automorphism of a totally disconnected, locally compact group G, then

Uα:={gG:αn(g)1 as n}

is a subgroup of G called the associated contraction group. In general, Uα need not be closed. However, if α is expansive, then the topology on Uαcan be made locally compact, i.e., it can be refined to a totally disconnected, locally compact group topology τ* with respect to which α|Uα is contractive (see [25, Proposition 9] for this fact, or our Lemma 2.3). In this way, the structure theory of locally compact contraction groups (see [24, 26, 10]) becomes available. In particular, the set

Tα:=tor(Uα)

of all torsion elements in the group Uα and the set

Dα:=div(Uα)

of divisible elements are α-stable closed subgroups of (Uα,τ*), and

(Uα,τ*)=Dα×Tα

internally as a topological group, if we endow Dα and Tα with the topology induced by (Uα,τ*) (see [10, Theorem B]).

Recall that the closure Uα¯ of the contraction group Uα in G plays a role in the structure theory of totally disconnected, locally compact groups; for example, the scale s(α-1) can be calculated as the module of the restriction of α-1 to Uα¯ (see [1, Proposition 3.21 (3)]). Our next theorem provides information on Uα¯, and on the divisible part Dα of Uα.

Let G be a totally disconnected, locally compact group and let α:GG be an automorphism such that Uα can be made locally compact (for example, any expansive automorphism). Then Uα¯=Dα×Tα¯ (internally) as a topological group, and Tα¯=Tαnub(α). In particular, Dα is an α-stable closed subgroup of G, and both the closure Tα¯ of Tα in G and nub(α) centralize Dα.

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 Uα and Uα-1 normalize nub(α).

Classes of examples are also considered. An analytic automorphism α of a Lie group G over a totally disconnected local field 𝕂 is expansive if and only if the associated Lie algebra automorphism L(α) is expansive, which means that none of its eigenvalues in an algebraic closure has absolute value 1. The Lie algebra L(G) of 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 GGLn(Qp) for some nN. Let α:GG be an expansive automorphism. Then G has an open α-stable subgroup which is nilpotent.

If α:GG is an expansive automorphism of a totally disconnected, locally compact group G, then UαUα-1 is an open subset of 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, UαUα-1 happens to be a subgroup of G (for instance, for all expansive automorphisms of closed subgroups GGLn(p), see Proposition 7.8). However, this is not always so, as can be seen from Remark 7.7. We also consider the localized completion Gp,q of a Baumslag–Solitar group

BS(p,q)=a,ttapt-1=aq

with primes pq, as recently studied in [4]. Then BS(p,q)Gp,q. We show the following theorem.

Let α:Gp,qGp,q be the conjugation by t. Then α is expansive but UαUα-1 is not a subgroup of Gp,q.

1 Preliminaries and basic facts

We write ={1,2,}, 0:={0} and :=0-. If J is a finite set, we let #J be its cardinality. We write XY for inclusion of sets, while XY means that X is a proper subset of Y. As usual, we write NG if 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 α-1 are continuous; similarly, both α and α-1 are assumed analytic if α is an automorphism of an analytic Lie group over a local field. We write Aut(G) for the group of all automorphisms of a topological group G. If a subgroup NG is stable under all αAut(G), then N is called topologically characteristic. A topological group G is called topologically perfect if its commutator group [G,G] is dense in G. If F is a finite group and X a set, we write FX:=xXF for the direct power endowed with the compact product topology. By contrast, F(X)FX is the subgroup of all (gx)xXFX such that gx=1 for all but finitely many xX. We shall always endow F(X) with the discrete topology. Surjective, open, continuous homomorphisms between topological groups are called quotient morphisms. A topological space X is called σ-compact if it is a union X=nKn of a sequence of compact sets KnX.

If α:GG is an automorphism of a locally compact group, choose a Haar measure λ on G and denote the module of α by ΔG(α); thus

ΔG(α)=λ(α(K))/λ(K)

for any compact subset KG with non-empty interior. If α:GG is an automorphism of a totally disconnected, locally compact group, we define nub(α), the contraction group Uα and its subgroups Tα and Dα as explained in the introduction. If VG is a compact open subgroup, we shall use the subgroups V+, V++, V- and V-- defined there, and abbreviate

V0:=V+V-=nαn(V).

We shall also need the so-called Levi factor

Mα:={gG:{αn(g):n} is relatively compact};

it is known that Mα is an α-stable closed subgroup of 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:

  1. (a)G is metrizable and has an α -stable, σ -compact open subgroup,
  2. (b)V--=Uα and V++=Uα-1 for each compact open subgroup VG such that V0={1},
  3. (c)G has a compact open subgroup V such that V=V+V- and V0={1},
  4. (d)UαUα-1 is open in G,
  5. (e)α|H is expansive, for each α -stable subgroup HG.

Proof.

(a) Since α is expansive, there exists an identity neighbourhood V such that nαn(V)={1}. After replacing V with a smaller compact identity neighbourhood if necessary, we may assume that V is compact. Since V is compact and (k=-nnαk(V))n a decreasing sequence of closed identity neighbourhoods in V with intersection {1}, the members of the sequence form a basis of identity neighbourhoods. Hence G is metrizable. The subgroup of G generated by nαn(V) is α-stable, open and σ-compact.

(b) By [1, Proposition 3.16], we have V--=UαV0 and thus V--=Uα. Likewise, V++=Uα-1.

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

nαn(W)={1}.

By [29, Lemma 1], there exists m such that V:=k=1mαk(W) satisfies V=V+V-. Then we have V0W0={1} and thus V0={1}, proving (c). The latter entails Uα=V-- and Uα-1=V++, by (b). In particular, V-Uα and V+Uα-1, entailing that V=V+V-UαUα-1. Thus UαUα-1 is an identity neighbourhood. Given gUα and hUα-1, the map GG given by xgxh is a homeomorphism which takes UαUα-1 onto itself and 1 to gh. Hence UαUα-1 has gh in its interior and thus UαUα-1 is open.

(e) If VG is an identity neighbourhood with nαn(V)={1}, then VH is an identity neighbourhood in H and nαn(VH)={1}. ∎

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:

  1. (a)The closure of Uα in G is Uα¯=Uαnub(α) (see [1, Corollary 3.30] if G is metrizable; the general case follows with [13]).
  2. (b)nub(α)=Uα¯Uα-1¯ (see [1, Corollary 3.27] if G is metrizable; the general case follows with [13]).
  3. (c)Both nub(α)Uα and nub(α)Uα-1 are dense in nub(α) (see [31, Theorem 4.1 (v) and Proposition 5.4 (i)]).
  4. (d)If NG is a closed α-stable normal subgroup, q:GG/N the canonical quotient morphism and α¯ the automorphism of G/N induced by α, then q(Uα)=Uα¯ (see [1, Theorem 3.8] if G is metrizable, [13] in the general case).
  5. (e)Uα is closed if and only if Uα-1 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 [13]).
  6. (f)The so-called parabolic subgroup
    Pα:={gG:{αn(g):n0} is relatively compact}
    normalizes Uα (see [1, Proposition 3.4]). Hence also its subgroups Mα=PαPα-1 and nub(α)Mα normalize Uα.

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:

  1. (a) α is expansive if and only if its restriction α|Mα to the Levi factor is expansive.
  2. (b)If Uα is closed, then α is expansive if and only if UαUα-1 is open in G, if and only if Mα is discrete.

Proof.

(a) In view of Lemma 1.1 (e), we only need to show that if α|Mα is expansive, then so is α. Let PMα be a compact, open identity neighbourhood such that nαn(P)={1}. There is a compact identity neighbourhood QG such that QMα=P. If gI:=nαn(Q), then αn(g)Q for each n, whence α(g) is relatively compact and thus gMα. Hence IMα. Since I is α-stable and IQMα=P, we deduce that

I=nαn(I)nαn(P)={1}.

Thus nαn(Q)={1} and thus α is expansive.

(b) If α is expansive, then UαUα-1 is open by Lemma 1.1 (d). If Uα is closed, then the set UαMαUα-1 is open in G and the product map

Uα×Mα×Uα-1UαMαUα-1,(x,y,z)xyz

is a homeomorphism (by 1.2 (e) above and part (f) from the theorem in [5]). Hence UαUα-1Mα={1}. If UαUα-1 is open, this implies that Mα is discrete. If Mα is discrete, then α|Mα is expansive and hence also α, by (a). ∎

If α is an automorphism of a totally disconnected, compact group G, then Uα and Uα-1 are normal in G (since G=Mα=Mα-1 in 1.2 (f)), and hence so is nub(α)=Uα¯Uα-1¯.

Recall that a group G is called a torsion group of finite exponent if there exists m such that gm=1 for all gG. If such a group is a subgroup of a topological group H, then also gm=1 for all g in the closure G¯ of G in H, and thus also G¯ is a torsion group of finite exponent.

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

  1. (a)nub(α) is open in G (see [31, Lemma 5.1]), whence also Uα¯ and Uα-1¯ are open in G (by 1.2 (b)).
  2. (b)G is a torsion group of finite exponent. [By (a) and 1.4, nub(α) is a normal subgroup and G/nub(α) is finite, hence a torsion group of finite exponent. It therefore suffices to show that nub(α) is a torsion group of finite exponent. This is immediate from [31, Proposition 4.4, Theorem 6.2, Proposition 6.3]).
  3. (c)Since Uα and Uα-1 are normal in G, the set UαUα-1 is an open α-stable subgroup of G contained in nub(α). Thus nub(α)=UαUα-1 by [31, Corollary 4.3].

Let α be an automorphism of a totally disconnected, locally compact group G. Let HG be a closed, α-stable subgroup. Then the following holds:

  1. (a)Then nub of α|H is contained in nub(α).
  2. (b)If nub(α)H, then nub(α|H)=nub(α).

Proof.

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

nub(α|H)=Uα|H¯U(α|H)-1¯Uα¯Uα-1¯=nub(α).

(b) Since nub(α)H, we have Uαnub(α)UαH=Uα|H and thus

nub(α)=nub(α)Uα¯Uα|H¯

(using 1.2 (c)). Likewise, nub(α)U(α|H)-1¯. Hence

nub(α)Uα|H¯U(α|H)-1¯=nub(α|H).

Since nub(α|H)nub(α) by (a), equality follows. ∎

We shall also need certain facts concerning contractive automorphisms.

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

  1. (a)If G{1}, then G is infinite and non-discrete. [If xG{1}, then we have αn(x)1 for all n and αn(x)1, entailing that the topological group G is not discrete and hence infinite.]
  2. (b)If G is locally compact, then α is compactly contractive, i.e., for each identity neighbourhood UG and compact set KG there exists m such that αn(K)U for all nm (see [24, Lemma 1.4 (iv)]). Moreover, G is non-compact (unless G={1}); see [24, Section 3.1].

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

  1. (a)If G{1}, then ΔG(α-1) is an integer 2 (see [10, Proposition 1.1 (e)]).
  2. (b)If G=G0G1G2Gn={1} is a series of α-stable closed subgroups of G such that Gj is a proper normal subgroup of Gj-1 for all j{1,,n}, then n is bounded by the number of prime factors of ΔG(α-1) (see [10, Lemma 3.5]).
  3. (c)G=div(G)×tor(G) as a topological group for a divisible torsion free group div(G) and a torsion group tor(G) of finite exponent (cf. [10, Theorem B]).
  4. (d)If NG is an α-stable closed normal subgroup, q:GG/N the canonical quotient morphism and α¯ the automorphism of G/N induced by α, then
    q(div(G))=div(G/N)andq(tor(G))=tor(G/N).
    [The inclusions q(div(G))div(G/N) and q(tor(G))tor(G/N) are clear. Since we have G/N=q(div(G)tor(G))=q(div(G))q(tor(G)) and G/N=div(G/N)×tor(G/N), equality follows.]

If an automorphism α:GG of a totally disconnected, locally compact group is contractive, then it is also expansive. To see this, let VG be any compact neighbourhood of the identity. If gG{1}, then G{g} is an identity neighbourhood, whence there exists n such that αn(V)G{g} (see 1.8 (b)). Thus gαn(V) and we have shown that nαn(V)={1}.

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

α:(G,τ)(G,τ)

be a contractive automorphism. We say that (G,α) (or simply G) can be made locally compact if there exists a locally compact topology τ* on G making it a topological group such that ττ* and α:(G,τ*)(G,τ*) is a contractive automorphism. We write G* for G, endowed with the topology τ*.

If τ is totally disconnected, then also τ* is totally disconnected (as the inclusion map G*G is continuous).

The topology τ* is unique (if it exists): see [25, Corollary 8] for a discussion in a more general framework. We give a streamlined proof in our setting.

The topology τ* is uniquely determined by the properties from Definition 2.1.

Proof.

Assume that τ^ is a group topology on G with the same properties as τ*. We show that the identity map

ϕ:(G,τ^)(G,τ*),xx

is continuous. Reversing the roles of τ^ and τ*, also ϕ-1 will be continuous and thus τ^=τ*. Because ϕ is a homomorphism, we need only prove its continuity at 1. By local compactness, there exists a compact identity neighbourhood V(G,τ^). After replacing V with the closure of its interior V0 in (G,τ^), we may assume that V0 is dense in V. Then V is also compact in (G,τ). Let U(G,τ*) be an arbitrary identity neighbourhood, and let W(G,τ*) be a compact identity neighbourhood such that WW-1U. Then W is also compact in (G,τ), hence closed in (G,τ) and hence closed in (G,τ^). Since α:(G,τ*)(G,τ*) is contractive, we have G=n0α-n(W). Hence V is the countable union of the closed subsets Vα-n(W), for n0. By the Baire Category Theorem, there exists m0 such that Vα-m(W) has non-empty interior in V. Since V0 is dense in V, we deduce that α-n(W)V0 has non-empty interior in V0, and so W has non-empty interior W0 in (G,τ^). Then W0(W0)-1 is an identity neighbourhood in (G,τ^) and hence U is an identity neighbourhood in (G,τ^). Since U=ϕ-1(U), we see that ϕ is continuous at 1. ∎

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

Let G be a totally disconnected, locally compact group. If an automorphism α:GG is expansive, then (Uα,α|Uα) and (Uα-1,α-1|Uα-1) can be made locally compact.

Proof.

Let VG be a compact open subgroup such that nαn(V)={1}. Then αn(V-)=V-k=1nαk(V) is open in V- for each n0. Since V- is compact and n0αn(V-)=nαn(V)={1}, it follows that the open subgroups αn(V-) form a basis of identity neighbourhoods in V-, for n0. If n and gUα=V--=k0α-k(V-) (see Proposition 1.1 (c)), then gα-k(V-) for some k0 and αn(V-) is an open subgroup of the topological group α-k(V-). Hence, there exists m0 such that gαm(V-)g-1αn(V-). By the preceding, there exists a group topology τ* on Uα for which the set {αn(V-):n0} is a basis of identity neighbourhoods. Thus V- is an open subgroup of (Uα,τ*), and the latter group induces the given compact topology on V- (as {αn(V-):n0} is a basis of identity neighbourhoods for both topologies on the subgroup V-). Thus (Uα,τ*) is a totally disconnected, locally compact group and α is still continuous (being continuous on the open subgroup V-) as well as α-1 (being continuous on the subgroup α(V-) which is open in V-). Since Uα=n0α-n(V-) and (αn(V-))n0 is a basis of identity neighbourhoods, it readily follows that the automorphism α of (Uα,τ*) is contractive. ∎

The group Uα can also be made locally compact if α is an arbitrary (not necessarily expansive) analytic automorphism of a Lie 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 G:=(p), where p is the additive group of p-adic integers. The contraction group Uα is non-trivial, as it is the group of all (zn)n such that zn0 as n-. Then Uα cannot be locally compact, because tor(Uα)tor(G)={0} and div(Uα)div(G)={0}. Thus, if Uα could be made locally compact, then we would have Uα=Dα+Tα={0} (using 1.9 (c)), contrary to Uα{0}.

Let G be a topological group and α:GG a contractive automorphism such that (G,α) can be made locally compact. Then we have:

  1. (a)(H,α|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*.
  2. (b)If ϕ:GH is a continuous homomorphism to a topological group H admitting an automorphism β:HH such that βϕ=ϕα, then β|ϕ(G) is contractive, ϕ(G) can be made locally compact and
    G*ϕ(G)*,xϕ(x)
    is a topological quotient map.

Proof.

(a) Let τ and τ* be as in Definition 2.1. Since ττ*, H is closed in G* (in either case) and hence H is a locally compact group in the topology σ on H induced by G*, which is finer than the topology induced by G and turns α|H into a contractive automorphism of (H,σ). Thus H*=(H,σ).

(b) Let σ be the topology on ϕ(G) turning G*(ϕ(G),σ), xϕ(x) into a quotient map. Then σ is finer than the topology induced on ϕ(G) by H. Moreover, (ϕ(G),σ)G*/kerϕ is locally compact. Since

β|ϕ(G)ϕ=ϕα:G*(ϕ(G),σ)

is continuous, the map β|ϕ(G) is continuous with respect to the quotient topology σ. Also (β|ϕ(G))-1 is continuous, by an analogous argument. Finally, the map β|ϕ(G):(ϕ(G),σ)(ϕ(G),σ) is contractive: Since βnϕ=ϕαn, this follows from the facts that α:G*G* is contractive and ϕ:G*ϕ(G)* is continuous. ∎

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

G=G0G1Gn

be α-stable closed subgroups of G such that Gj is normal in Gj-1 for all indices j{1,,n}. Let J be the set of all j{1,,n} such that Gj-1/Gj is not discrete. Then

#Jα+α-1,

where α is the number of prime factors of ΔUα*(α-1|Uα*) and α-1 is the number of prime factors of ΔUα-1*(α|Uα-1*).

Proof.

Let Jα (resp., Jα-1) be the set of all j{1,,n} such that

UαGjUαGj-1

(resp., Uα-1GjUα-1Gj-1). If j{1,,n}(JαJα-1), then

UαGj=UαGj-1andUα-1Gj=Uα-1Gj-1.

Since α is expansive, it follows that (UαGj-1)(Uα-1Gj-1) is open in Gj-1 (see Lemma 1.1 (d) and (e)). We deduce that Gj is open in Gj-1 and thus jJ. Hence JJαJα-1, whence #J#Jα+#Jα-1α+α-1 (using 1.9 (b) in the last step). ∎

We shall use a simple fact.

Let K be a compact group and DK a subgroup which is divisible. Then also the closure D¯ is divisible. If K is totally disconnected, then D={1}.

Proof.

For each m, the map fm:D¯D¯ given by ggm is continuous and hence has compact image. As the image contains D by hypothesis, we see that fm(D¯)=D¯. Thus D¯ is divisible.

If K is totally disconnected, then K is a pro-finite group. In particular, the homomorphisms f:KF to finite groups F separate points on D¯. But each f(D¯) is both finite and divisible and therefore the trivial group. Hence also D¯={1}. ∎

Let α be an automorphism of a totally disconnected, locally compact group G. If Uα can be made locally compact (e.g., if α is expansive), then the following holds:

  1. (a)Uαnub(α)=Tαnub(α);
  2. (b)nub(α)=Tαnub(α)¯;
  3. (c)Tα¯Uα=Tα;
  4. (d)Tα¯=Tαnub(α).

If both Uα and Uα-1 can be made locally compact, then we also have:

  1. (e)nub(α)=Tα¯Tα-1¯.

Proof.

(a) By Lemma 2.6 (a), Uαnub(α) can be made locally compact. Thus α restricts to a contractive automorphism β of (Uαnub(α))*, enabling us to write (Uαnub(α))*=DβTβ. Since nub(α) is compact and totally disconnected, its divisible subgroup Dβ has to be trivial, by Lemma 2.8. Thus

Uαnub(α)=Uβ=TβTα

and hence Uαnub(α)=Tαnub(α).

(b) As nub(α)=Uαnub(α)¯ (see 1.2 (c)), the assertion is immediate from (a).

(c) The subgroup TαTα¯Uα is trivial. Because Tα is a torsion group of finite exponent (see 1.9 (c)), also Tα¯ is a torsion group, see 1.5. Let β the restriction of α to the closed α-stable subgroup Tα¯Uα* of Uα*. Then Tα¯Uα=DβTβ. Since Dβ is torsion-free (see 1.9 (c)) and Tα¯ a torsion group, Dβ={1} follows. Thus

Tα¯Uα=TβTα.

(d) We have Tα¯Uα¯=Uαnub(α) (see 1.2 (a)). Since nub(α)Tα¯ by (b), we deduce that

Tα¯=(UαTα¯)nub(α)=Tαnub(α)

(using (c) for the last equality).

(e) We have

nub(α)=Uα¯Uα-1¯Tα¯Tα-1¯

(see 1.2 (b)) and

nub(α)=Tαnub(α)¯Tα-1nub(α)¯Tα¯Tα-1¯

(using (b)). ∎

Let α be an automorphism of a locally compact group G and let HG be an α-stable subgroup such that α|H is contractive. Then the closure H¯G is σ-compact.

Proof.

Let KH¯ be a compact identity neighbourhood. Then nαn(K) is a σ-compact subset of H¯ and generates a σ-compact subgroup S of H¯. Since S is an α-stable open subgroup of H¯ and α|H is contractive, we have HS and thus S=H¯ (since S is closed). Hence H¯=S is σ-compact. ∎

3 Proof of Theorem A

We now prove Theorem A.

Proof of Theorem A.

Let G be a totally disconnected, locally compact group, α an expansive automorphism of G and NG an α-stable, closed normal subgroup. Let q:GG/N be the canonical quotient morphism and α¯ the automorphism of G/N induced by α (determined by α¯q=qα). Then α|N is expansive, by Lemma 1.1 (e). We show that α¯ is also expansive. By Proposition 1.3 (a), we need only show that α¯ restricts to an expansive automorphism of Mα¯. After replacing G with q-1(Mα¯) (which is closed since Mα¯ is closed), we may assume that G/N=Mα¯. Let U be a subgroup of G/N tidy for α¯. Then U=U+=U- as α¯(g) is relatively compact for each gU (cf. [29, Lemma 9]) and thus U is an α¯-stable, compact open subgroup of G/N. After replacing G with q-1(U), we may assume that G/N is compact. Using [31, Proposition 5.1] and the metrizability of G/N, we find a descending sequence (Hn)n of α¯-stable closed normal subgroups Hn of G/N such that α¯ induces an expansive automorphism αn on (G/N)/Hn for each n and G/N is the projective limit

G/N=lim(G/N)/Hn.

Set Ln:=q-1(Hn); then (Ln)n is a descending sequence of α-stable closed normal subgroups of G, with nLn=N.

There exists m such that Ln is open in Lm for all nm. Indeed, if this were false, we could find a subsequence (Lnk)k such that, for each k, the normal subgroup Lnk+1 is not open in Lnk. This contradicts Proposition 2.7.

After passing to a subsequence, we may assume that Ln is open in L1 for each n. Hence Ln contains both UαL1 and Uα-1L1. As a consequence, N=nLn contains both UαL1 and Uα-1L1. Hence N is open in L1 (and in each Ln), using the fact that α|L1 is expansive and thus (UαL1)(Uα-1L1) an open subset of L1 (see 1.1 (d) and (e)). This implies that the compact group H1L1/N is discrete and hence a finite group. Since H1H2 with nHn={1}, we deduce that Hn={1} for some n. Since α¯ corresponds to the expansive automorphism αn on (G/N)/HnG/N, we see that α¯ is expansive.

Conversely, assume that both α|N and α¯:G/NG/N are expansive.1 Then there is an open identity neighbourhood PG/N such that nα¯n(P)={1}, and an open identity neighbourhood QN such that nαn(Q)={1}. After shrinking Q, we may assume that Qq-1(P). Then Q=NV for some open identity neighbourhood VG. After replacing V with Vq-1(P), we may assume that Vq-1(P). We have

N=q-1(nα¯n(P))=nαn(q-1(P)),

entailing that I:=nαn(V) is an α-stable subset of N. Since

I=INVN=Q,

we deduce that I=nαn(I)nαn(Q)={1}. Hence I={1} and α is expansive.∎

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 NG is open or discrete. Let CG be the topologically characteristic subgroup2 of G generated by UαUα-1. Then C is an open normal subgroup of G, and one of the following cases occurs:

  1. (a)[C,C]¯ is open in G, in which case C=[C,C]¯ is topologically perfect.
  2. (b)[C,C]¯ is discrete.

Proof.

Note that C is an open subgroup of G as it contains the open set UαUα-1 (see Lemma 1.1 (d)). The closed subgroup [C,C]¯ is topologically characteristic in C, whence it is topologically characteristic in G and hence α-stable and normal. Therefore [C,C]¯ is open or discrete. If [C,C]¯ is open, then it contains UαUα-1. Hence [C,C]¯=C, using the fact that C is the smallest topologically characteristic subgroup of G which contains UαUα-1. ∎

Proof of Theorem B.

Define α and α-1 as in Proposition 2.7. For every series

Σ:G=G0G1Gn={1}

of α-stable closed subgroups of G, let JΣ be the set of all j{1,,n} such that Gj-1/Gj is not discrete. Then JΣα+α-1, by Proposition 2.7, entailing that the maximum

m:=maxΣJΣ

over all series Σ exists. Let Σ:G=G0G1Gn={1} be a series with JΣ=m. Let NGj-1 be an α-stable closed normal subgroup with GjN. If neither Gj-1/N nor N/Gj were discrete, we would have JΣ{N}=JΣ+1, a contradiction. Thus N will be open in Gj-1 or Gj is open in N.

For an index jJΣ, let qj:Gj-1Gj-1/Gj be the canonical quotient morphism, let αj:Gj-1/GjGj-1/Gj be the automorphism induced by α and let CjGj-1/Gj be the topologically characteristic subgroup generated by UαjUαj-1. If [Cj,Cj]¯ is open in Gj-1/Gj, define

Mj:=Nj:=qj-1(Cj);

thus we have that Gj-1/Mj(Gj-1/Gj)/Cj and Mj/Nj={1} are discrete and Nj/GjCj is topologically perfect. If [Cj,Cj]¯ is discrete, we define

Mj:=qj-1(Cj)andNj:=qj-1([Cj,Cj]¯)

(by Lemma 4.1, only these two cases can occur). So Gj-1/Mj(Gj-1/Gj)/Cj is discrete, Mj/NjCj/[Cj,Cj]¯ is abelian and Nj/Gj[Cj,Cj]¯ is discrete. Hence

Σ:=ΣjJΣ{Mj,Nj}

is a series of α-stable closed subnormal subgroups such that all non-discrete subfactors are abelian or topologically perfect. Since #JΣ=#JΣ is maximal, all non-discrete subfactors of Σ have the property that all stable closed normal subgroups are open or discrete.∎

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 {1}, then (G,α) is called a 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 (G,α) is a simple expansion group. We remark:

If G is second countable in Theorem B, then one can achieve there that each of the quotient groups Gj-1/Gj is discrete, abelian, both topologically perfect and elementary, or an elementary-free simple expansion group.

In fact, let us consider a topologically perfect factor Q:=Gj-1/Gj in a series all of whose factors are discrete, abelian, or topologically perfect, and which has a maximum number of non-discrete factors. By [28, Theorem 7.15], there are two topologically characteristic, closed subgroups D1 and D2 of Q such that QD1D2{1} and, moreover, both D2 and Q/D1 are elementary and D1/D2 is elementary-free. Let N1 and N2 be the pre-images of D1 and D2, respectively, under the quotient morphism Gj-1Gj-1/Gj. Then N1 and N2 are α-stable closed normal subgroups of Gj-1, and Gj-1N1N2Gj.

Case 1: If D1/D2 is non-trivial, the elementary-free group N1/N2D1/D2 is non-discrete (as it would be elementary otherwise), whence N2 is not open in N1. Hence N2 is not open in Gj-1 and hence N2/Gj is discrete (by maximality of the number of non-discrete factors). Again by maximality, Gj-1/N1 is discrete and N1/N2 does not have closed normal subgroups stable under the induced expansive automorphism other than open or discrete subgroups. So, the elementary-free group N1/N2 is a simple expansion group.

Case 2: If D1/D2 is trivial, then the topologically perfect group Gj-1/Gj=QD1=D2{1} is elementary, as any extension of elementary groups is.

5 Proof of Theorem C

We prove Theorem C and record some related results.

Proof of Theorem C.

Since nub(α)Uα¯, we may replace G with Uα¯ without changing the nub (see 1.7), or Tα¯, or Dα. We may therefore assume that Uα is dense in G. Since nub(α) normalizes Uα (see 1.2 (f)), Uα is a normal subgroup of G=Uαnub(α) (exploiting 1.2 (a)). Hence also the characteristic subgroups Dα and Tα of Uα are normal in G. Therefore also Tα¯ is normal in G. Since Tα¯ is a torsion group (see 1.9 (c) and 1.5) and Dα torsion-free (see 1.9 (c)), we see that DαTα¯={1}. Moreover, using the fact that Tαnub(α)=Tα¯ by Lemma 2.9 (d), we obtain

G=Uαnub(α)=DαTαnub(α)=DαTα¯.

Hence G=Dα×Tα¯ as an abstract group. In particular, Dα centralizes Tα¯. Thus Dα also centralizes nub(α)Tα¯. Since Tα¯ is σ-compact by Lemma 2.10, also Dα*×Tα¯ is a σ-compact locally compact group (writing Dα* for Dα, endowed with the locally compact topology induced by Uα*). Because also G is locally compact and the product map π:Dα*×Tα¯G, (x,y)xy is a continuous isomorphism of abstract groups, we deduce from [11, Section 5.29] that π is an isomorphism of topological groups. Hence G=Dα×Tα¯ (internally). In particular, Dα is closed in G.∎

Let G be a totally disconnected, locally compact group and α an automorphism of G such that Uα and Uα-1 can be made locally compact (e.g., any expansive automorphism). Then DαDα-1={1}.

Proof.

Since Dα and Dα-1 are closed and α-stable, it follows that their intersection H:=DαDα-1 is a totally disconnected, locally compact contraction group for both α|H and α-1|H. Hence H={1}. Indeed, if H{1}, then both ΔH(α|H) and ΔH(α|H-1)=ΔH(α|H)-1 would be integers 2 (see 1.9 (a)), which is impossible. ∎

The nub of an expansive automorphism α:GG need not have an open normalizer in G. To see this, let F be a finite group which is a semidirect product F=NH of a normal subgroup N and a subgroup H which is not normal in F (e.g., F might be the dihedral group C3C2). Let G be the group of all (nk,hk)kF such that (nk)kN(-)×N0=:M. Thus G=MH as an abstract group. Endow G with the topology making it the direct product topological space of the restricted product M and the compact group H. Then G is a topological group, being the ascending union of the open subgroups H{k:k<-m}×F{k:k-m} for m, which are topological groups. The right-shift α is an automorphism of G. We have

Uα=M(H(-)×H0)

and

Uα-1=H-×H(0).

Thus Uα¯=G, Uα-1¯=H and nub(α)=Uα¯Uα-1¯=H (using 1.2 (b)). As H is not normal in F, we see that nub(α)=H is not normal in G. If the normalizer NG(nub(α)) was open in G, then (being α-stable), it would contain the dense subgroup Uα of G and hence coincide with G (a contradiction). Thus NG(nub(α)) is not open.

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 j{1,,n} such that Gj-1/Gj is abelian and non-discrete. Let αj be the automorphism of Gj-1/Gj induced by α and let qj:Gj-1Gj-1/Gj be the quotient homomorphism, for jI. Then UαjUαj-1 is an open αj-stable subgroup of Gj-1/Gj and hence Hj:=qj-1(UαjUαj-1) is an α-stable open normal subgroup of Gj-1. Then Gj-1/Hj is discrete and all stable, closed, proper subgroups of Hj/Gj are discrete. After inserting the Hj into the series for all jI, we may thus assume without loss of generality that all abelian, non-discrete subfactors Gj-1/Gj have the property that all of their αj-stable, closed, proper subgroups are discrete, and that Gj-1/Gj=UαjUαj-1.

Let Gj be a topological group and αjAut(Gj) for j{1,2}. We say that (G1,α1) and (G2,α2) are isomorphic if there exists an isomorphism ϕ:G1G2 of topological groups such that α2ϕ=ϕα1.

Let A{1} be an abelian, totally disconnected, locally compact group and let α:AA be an expansive automorphism. Assume that A=UαUα-1 and assume that every α-stable proper closed subgroup of A is discrete. Then there exists a prime number p such that (A,α) isomorphic to one of the following:

  1. (a)pn for some n, together with a contractive linear automorphism β:pnpn not admitting non-trivial proper β -stable vector subspaces,
  2. (b)pn for some n, together with β-1 for a contractive linear automorphism β:pnpn not admitting non-trivial proper β -stable vector subspaces,
  3. (c)Cp(-)×Cp0 with the right-shift,
  4. (d)Cp(-)×Cp0 with the left-shift,
  5. (e)Cp with the right-shift.

Proof.

Let Dα be the divisible part and let Tα be the torsion part of Uα, and define Dα-1 and Tα-1 analogously. If Dα{1}, then Dα=Dα* is an α-stable closed subgroup (see Theorem C) which is non-discrete (see 1.8 (a)) and thus A=Dα. By 1.8 (a) and the hypotheses, Dα is a divisible simple contraction group and hence of the form described in (a) (see [10, Theorem A]). Likewise, A is of the form described in (b) whenever Dα-1{1}.

Throughout the rest of the proof, assume that Dα=Dα-1={1}. Then we have A=UαUα-1=TαTα-1.

Since nub(α) is an α-stable closed subgroup of A, it is either all of A or discrete. Being also compact, it is finite in the latter case, and thus {1} is an open α-stable (normal) subgroup of nub(α). Now [31, Corollary 4.4] (proper such do not exist) shows that nub(α)={1}.

Case nub(α)={1}: In this case Tα=Uα=Uαnub(α) and Tα-1=Uα-1=Uα-1nub(α) are closed α-stable subgroups of A (using 1.2 (a)). If Tα{1}, then Tα is non-discrete. Hence Tα=A by the hypotheses, and this is a simple contraction group which is a torsion group and hence of the form described in (c) (see [10, Theorem A]). Likewise, A is of the form described in (d) if Tα-1{1}.

Case A=nub(α): In this 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 Uα or Uα-1 is non-trivial and hence non-discrete, being a contraction group). Hence, by [31, Proposition 6.3], (A,α) is isomorphic to the right-shift of F for a finite simple group F. Since A is abelian, FCp for some p and thus A is of the form described in (e). ∎

Let G be a totally disconnected, locally compact group and let α:GG be an expansive automorphism. If G is abelian, then the map

π:Uα*×Uα-1*G,(x,y)xy

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 Uα and Uα-1, respectively.

[To see this, let VG be a compact open subgroup such that V+V-={1} and V=V+V- (see Lemma 1.1 (c)). Then V- and V+ are open subgroups of Uα* and Uα-1*, respectively (see proof of Lemma 2.3). Then π(V+×V-)=V is open in G and π|V+×V- is injective, as vw=vw for v,vV+, w,wV- implies v-1v=w(w)-1V+V-={1} and thus v=v and w=w. Since V+×V- is compact, it follows that π restricts to a homeomorphism V+×V-V. Since π(gv,wh)=gπ(v,w)h for all gUα, hUα-1 and (v,w)V+×V-, also π|gV+×V-h is a homeomorphism onto an open set.]

It can happen that Uα is closed for an expansive automorphism α of a totally disconnected, locally compact group G, but Uα¯ is not closed for the induced automorphism α¯ on G/N for some α-stable closed normal subgroup NG. The following example also illustrates Remark 6.3.

Given a non-trivial finite abelian group (F,+), consider the restricted products

H1:=F(-)×F0andH2:=F-×F(0),

with V1:=F0 and V2:=F-, respectively, as compact open subgroups. Let α be the right-shift on G:=H1×H2 (i.e., on both H1 and H2). Then α is an automorphism and it is expansive as nαn(V1×V2)={0}. Moreover, Uα=H1 and Uα-1=H2 are closed. Also, let α¯ be the right-shift on F. Then

q:GF,(f,g)f+g

is a continuous surjective homomorphism. Restricted to the compact open subgroup V1×V2, the map q is an isomorphism of topological groups. Hence q is open, has discrete kernel, and is a quotient morphism. Finally, Uα¯=F(-)×F0 is a dense proper subgroup in F. Hence Uα¯ is not closed in FG/ker(q).

Another property can be observed.

Let G be a totally disconnected, locally compact group that is abelian, and α:GG be an expansive automorphism. Then the torsion subgroup tor(G) is closed in G.

Proof.

Since V:=UαUα-1 is an open subgroup of G, we need only show that Vtor(G)=tor(V) is closed. After replacing G with its α-stable subgroup V, we may therefore assume that G=UαUα-1. Since Dα and Dα-1 are torsion-free (see 1.9 (c)) and DαDα-1={1} by Corollary 5.1, we deduce that DαDα-1 is isomorphic to Dα×Dα-1 as an abstract group and hence torsion-free. Hence DαDα-1TαTα-1={1}. Combining this with

G=UαUα-1=DαDα-1TαTα-1,

we see that

G=(DαDα-1)×(TαTα-1)=Dα×Dα-1×TαTα-1

internally as an abstract group. By equation (6.1), the torsion subgroup of G is tor(G)=TαTα-1. Hence tor(G) has finite exponent (like Tα and Tα-1). Thus also tor(G)¯ is a torsion group (by 1.5) and thus tor(G)=tor(G)¯. ∎

7 Example: p-adic Lie groups

Let 𝕂 be a local field and let || be an absolute value on 𝕂 defining its topology (see [27]). We pick an algebraic closure 𝕂¯ containing 𝕂 and use the same symbol, ||, for the unique extension of the absolute value on 𝕂 to an absolute value on 𝕂¯ (see [20, Theorem 16.1]). If E is a finite-dimensional 𝕂-vector space and β:EE a 𝕂-linear automorphism, we write R(β) for the set of all absolute values |λ| of zeros λ of the characteristic polynomial of β in 𝕂¯. We let E~λE𝕂𝕂¯ be the generalized eigenspace of β𝕂id𝕂¯ for the eigenvalue λ. For ρR(β), we let

Eρ:=(|λ|=ρE~λ)E.

Then E=ρR(β)Eρ (see [17, Chapter II, Section 1]) and we recall that

E=UβMβUβ-1

with

Mβ=E1,Uβ=ρ<1EρandUβ-1=ρ>1Eρ

(cf. [8, Lemma 2.5]).

If G is a Lie group over 𝕂, then its tangent space L(G):=T1(G) at the identity element carries a natural Lie algebra structure, and L(α):L(G)L(H) is a Lie algebra homomorphism for each 𝕂-analytic homomorphism α:GH between 𝕂-analytic Lie groups. We abbreviate Ad(g):=L(Ig), where Ig:GG is given by xgxg-1 for gG (cf. [23] for further information).

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

  1. (a) α is expansive.
  2. (b)β:=L(α):L(G)L(G) is expansive.
  3. (c)1R(β).

If α is expansive, then L(G) is a nilpotent Lie algebra.

Proof.

(a)  (b) By contraposition. If (b) is false, then β is not expansive. To deduce that α is not expansive, let VG be an identity neighbourhood. Since Uβ is a vector subspace of L(G) by (7.1) and hence closed, using Proposition 1.3 (b) we deduce that Mβ is not discrete and hence a non-trivial vector subspace (in view of (7.1)). But then 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 T1(W)=Mα (whence W{1}); see Proposition 6.3 (a) and part (b) of the Local Invariant Manifold Theorem in [8]; cf. also [7]. Then {1}Wnαn(V), and thus α is not expansive.

(b)  (c) Since Uβ is closed, β is expansive if and only if Mβ=L(G)1 is discrete. Since L(G)1 is a vector space, the latter holds if and only if L(G)1={0}, i.e., 1R(β).

(c)  (a) Note that Uα and Uα-1 are immersed Lie subgroups of G with Lie algebras Uβ and Uβ-1, respectively (see [8, Theorem D]). We write Uα* for Uα as a Lie group; because the underlying topology is locally compact and α restricts to a contractive Lie group automorphism, this is consistent with the definition of Uα* in Section 2. Likewise, we consider Uα-1* as a Lie group. If 1R(β), then L(G)=UβUβ-1, entailing that the product map π:Uα*×Uα-1*G given by (x,y)xy has invertible differential at (1,1) (the addition map Uβ×Uβ-1L(G)=UβUβ-1). Thus, by the inverse function theorem [23], there exist open identity neighbourhoods VUα* and WUα-1* such that VW is open in G and the restriction

π|V×W:V×WVW

is an analytic diffeomorphism. Using [24, Lemma 3.2 (i)], we find compact open subgroups PV of Uα* and QW of Uα-1* such that α(P)P, α-1(P)V, α-1(Q)Q and α(Q)W. Then PQ is an open identity neighbourhood in G and we now show that nαn(PQ)={1}. To this end, let xP and yQ. If x1, then α-n(x)P for some n, which we choose minimal. Thus α-n+1(x)P and hence α-n(x)V. Since α-n(x)VP by the preceding and α-n(y)Q, we see that α-n(xy)=α-n(x)α-n(y)(VP)Q. As the map (7.2) is a bijection, we deduce that α-n(xy)PQ. Likewise, αm(xy)PQ for some m if y1. Thus nαn(PQ)={1} indeed and thus α is expansive.

Final assertion. If (c) holds, then β is a Lie algebra automorphism of L(G) and |λ|1 for all eigenvalues λ of β𝕂id𝕂¯ in 𝕂¯, entailing that none of the λ is a root of unity. Hence L(G) is nilpotent (see [3, Exercise 21 (b) among the exercises for Section 4 of Part I] or [12, Theorem 2]). ∎

For each continuous homomorphism θ:pGLn(p), there exists a nilpotent n×n-matrix xpn×n such that θ(t)=exp(tx) for all tp, using the matrix exponential function [19, Theorem 1.1]. Thus θ(0)=x uniquely determines θ, and so does θ|W for any 0-neighbourhood Wp.

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

  1. (a)For each gG, there is a unique continuous homomorphism θg:pG such that θg(1)=g. Moreover, {θg(0):gG}=L(G).
  2. (b)If 𝔥L(G) is an L(α)-stable Lie subalgebra, then there exists an α -stable Lie subgroup H of G with L(H)=𝔥.

Proof.

(a) Let *:L(G)×L(G)L(G) be the Campbell–Hausdorff multiplication on the nilpotent Lie algebra L(G). Because (G,α) and ((L(G),*),L(α)) are locally isomorphic contraction groups, they are isomorphic (see [26, Proposition 2.2]). The nilpotent group (L(G),*) inherits unique divisibility from the group (L(G),+), since ng (in the vector space L(G)) coincides with gn (in (L(G),*)). It is clear from this that θg(t)=tg is the unique continuous homomorphism p(L(G),*) with θg(1)=g. It satisfies g=θg(0).

(b) We may work with the isomorphic group (L(G),*) instead of G. Now H:=𝔥 is an L(α)-stable Lie subgroup of (L(G),*) with Lie algebra 𝔥. ∎

Let G be a linear p-adic Lie group. Assume that G is generated by θΘθ(Qp) for a set Θ of continuous homomorphisms θ:QpG, and L(G) is generated by {θ(0):θΘ} as a Lie algebra. Then the centre of G coincides with the kernel of Ad:GAut(L(G)).

Proof.

Let gG. For each θΘ, the map Igθ:pG, tgθ(t)g-1 is a continuous homomorphism such that (Igθ)(0)=Ad(g)θ(0). Thus, by 7.2, Igθ=θ if and only if Ad(g)θ(0)=θ(0). Since θΘθ(p) generates G, we see that gZ(G) if and only if Ad(g)θ(0)=θ(0) for all θΘ. The latter is equivalent to Ad(g)(x)=x for all xL(G), since {xL(G):Ad(g)(x)=x} is a Lie subalgebra of L(G) and L(G) is generated by θ(0) for θΘ by hypothesis. ∎

Proof of Theorem D.

After replacing G with an open subgroup, we may assume that G is generated by UαUα-1 (see Lemma 1.1 (d)). We prove that G is nilpotent in this case, by induction on the dimension dim(G) of G as a p-adic manifold. If dim(G)=0, then G is discrete, whence Uα=Uα-1={1} and the group G=UαUα-1={1} is nilpotent.

Now assume that dim(G)>0. After replacing G with an isomorphic group, we may assume that G is a subgroup of GLn(p) for some n, and that the inclusion map GGLn(p) is continuous (but not necessarily a homeomorphism onto its image). Then L(G) is a non-zero nilpotent Lie algebra (see (a)  (d) in Proposition 7.1) and so it has centre Z(L(G)){0}. The centre is L(α)-stable, and the restriction β of L(α) to the centre is expansive (like L(α)). Hence Z(L(G))=UβUβ-1. After replacing α with α-1 if necessary, we may assume that Uβ-1{0}. According to Lemma 7.3 (b), there is an α-stable Lie subgroup HUα-1 with L(H)=Uβ-1. We claim that H is in the centre Z(G) of G. If this is true, then Z(G) has positive dimension. Thus G/Z(G) is a Lie group of dimension dim(G/Z(G))<dim(G), and it is a linear Lie group as it injects into Aut(L(G)), by Lemma 7.4.3 By induction, G/Z(G) is nilpotent and hence so is G.

To prove the claim, let hH and let θ:pHUα-1 be a continuous homomorphism with θ(1)=h (see Lemma 7.3 (a)). Then x:=θ(0)L(H)Z(L(G)), entailing that ad(x):=[x,]=0. Now θ(t)=exp(tx) for all tp, by 7.2. For |t| small, we have Ad(θ(t))=Ad(exp(tx))=etad(x)=idL(G), using [3, Chapter III, Section 4, no. 4, Corollary 3]). Thus Adθ=idL(G), by the uniqueness assertion of 7.2, applied to Adθ:pAut(L(G)). In particular, Ad(h)=Ad(θ(1))=idL(G) and thus hZ(G), by Lemma 7.4.∎

If G is a totally disconnected, locally compact group which is a nilpotent group, let {1}=Z0Z1Zn=G be its ascending central series defined recursively via Zk:=qk-1(Z(G/Zk-1)), where qk:GG/Zk-1 is the canonical quotient morphism. Let α be an expansive automorphism of G and αk the induced automorphism of Gk/Gk-1.

If Zk/Zk-1=UαkUαk-1 for all k{1,,n} in the situation of 7.5, then G=UαUα-1. In particular, UαUα-1 is a subgroup of G.

Proof.

If n=0, then G={1}=UαUα-1. If n1, let β be the expansive automorphism of G/Z(G) induced by α, and let q:GG/Z(G) be the canonical quotient morphism. Then Z1=Z(G)=(UαZ(G))(Uα-1Z(G)) by the hypotheses and G/Z(G)=UβUβ-1 by induction. Since

q(UαUα-1)=UβUβ-1=G/Z(G),

we have

G=UαUα-1Z(G)
=UαUα-1(UαZ(G))(Uα-1Z(G))
=Uα(UαZ(G))Uα-1(Uα-1Z(G))
=UαUα-1.

Note that we can easily achieve that G/Zn-1=UαnUαn-1 after replacing G with its open subgroup generated by UαUα-1. However, the hypotheses on Zk/Zk-1 for k<n cannot always be achieved by passing to an open subgroup (as the following example illustrates).

The following example shows that even for nilpotent p-adic Lie groups with an expansive automorphism α, the set UαUα-1 may fail to be a subgroup. The example also provides a 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 H=p3 be the 3-dimensional p-adic Heisenberg group whose binary operation is given by

(x1,y1,z1)(x2,y2,z2)=(x1+x2,y1+y2,z1+z2+x1y2)

for all (x1,y1,z1),(x2,y2,z2)H. Let N={(0,0,z)H:|z|1}. Then N is a compact central subgroup of H. Identify G=H/N with p×p×(p/p) as a set. Define α:GG by

α(x,y,z+p)=(px,p-1y,z+p)

for all (x,y,z+p)G. Then α is a continuous automorphism of the p-adic Lie group G with Mα={(0,0,z+p):zp}, Uα={(x,0,0):xp} and Uα-1={(0,y,0):yp}. Since Mα is discrete, it follows that α is an expansive automorphism. As [Uα,Uα-1]={(0,0,z+p):zp} and UαUα-1={(x,y,xy+p):x,yp}, we get that UαUα-1 is a not a subgroup.

Let G be a closed subgroup of GLn(Qp) and α an expansive automorphism of G. Then UαUα-1 is an open (unipotent algebraic) subgroup of G.

Proof.

Replacing G by the group generated by Uα and Uα-1, we may assume by Theorem E that G is a closed nilpotent subgroup of GLn(p). Let 𝔾 be the Zariski closure of G. Then 𝔾 is defined over p and 𝔾 is nilpotent (cf. [2, Proposition 1.3 (b) and Corollary 1 in Section 2.4]). Since Uα and Uα-1 consists of one-parameter (unipotent) subgroups, 𝔾 is Zariski-connected. This implies that the set of unipotent elements form a subgroup 𝔾u, known as the unipotent radical (cf. [2, Theorem 10.6]). Since Uα and Uα-1 consists of one-parameter (unipotent) subgroups, Uα,Uα-1𝔾u. This implies that 𝔾=𝔾u, that is 𝔾 is an unipotent algebraic group, hence 𝔾 is defined over p (cf. [2, Section 4.5] and the fact that p-closed and defined over p are same as characteristic of p is zero) and G𝔾(p).

For i1, let Di=[𝔾(p),Di-1] with D0=𝔾(p) and Gi=[G,Gi-1]¯ with G0=G. Then Dk+1 is trivial for some k1 as 𝔾 is unipotent and GiDi. Thus, Gk is a closed α-stable subgroup of Dk which is a vector space. Let V be the maximal vector subspace of Gk. Then V is a closed α-stable central subgroup of G. The automorphism β:Gk/VGk/V defined by β(x+V)=α(x)+V for xGk is expansive. Since V is the maximal vector subspace of Gk and Gk is a closed subgroup of the p-adic vector space Dk, we get that Gk/V is a compact subgroup of the p-adic vector space Dk/V. Since the automorphism group of a compact p-adic analytic group is compact, compact p-adic analytic groups do not admit expansive automorphisms unless finite, hence V=Gk. This implies that Gk=Dk and Gk=V=(UαV)(Uα-1V). Since G/Gk is a closed subgroup of 𝔾(p)/Dk which is a linear (p-adic algebraic) group, the result follows by induction. ∎

In the case of linear p-adic Lie groups, even if UαUα-1 is an open subgroup for an expansive automorphism, the following example shows that it is not possible to have either of Uα or Uα-1 to normalize the other.

Let H be the 3-dimensional p-adic Heisenberg group defined as in Remark 7.7. For i=1,2, define αi:HH by

α1(x,y,z)=(px,p-2y,p-1z),α2(x,y,z)=(p2x,p-1y,pz)

for (x,y,z)H. Let G=H×H and α=α1×α2. Then

Uα={(x,0,0):xp}×{(a,0,c):a,cp}

and

Uα-1={(0,y,z):y,zp}×{(0,b,0)bp}.

Thus UαUα-1=G. Since {(x,0,0):xp} and {(0,y,0):yp} are not normal subgroups of H, neither Uα or Uα-1 normalize the other.

8 Example: Baumslag–Solitar groups

Throughout this section, we fix primes pq. We let

BS(p,q):=a,ttapt-1=aq

be the Baumslag–Solitar group. Then agag-1 has finite index in a for each gBS(p,q), and ggag-1={1}, hence the Schlichting completion Gp,q of BS(p,q) can be formed, which is a certain totally disconnected, locally compact group in which BS(p,q) is dense, and in which K:=a¯ is a compact open subgroup (see [4], cf. [21, 9]). We are interested in the inner automorphism

α:Gp,qGp,q,xtxt-1.

Proof of Theorem E.

By [4, Proposition 8.1], K contains an open subgroup Vp×q and K/V is a cyclic group of order dividing gcd(p,q)=1. Thus K=Vp×q. After multiplication with a unit, we may assume that the isomorphism takes a to (1,1).

Let G=(p×q) be the semidirect product of and p×q given by

(n,u,v)(m,u,v)=(n+m,u+(q/p)nu,v+(q/p)nv)

for all n,m, u,up and v,vq. The isomorphism Kp×q gives a homomorphism from a to G. Since

(1,0,0)(0,p,p)(-1,0,0)=(0,q,q),

sending t(1,0,0) yields a group homomorphism ϕ:BS(p,q)G. Since ϕ|a is a continuous homomorphism, ϕ extends to a continuous homomorphism of Gp,q into G which would also be denoted by ϕ. Since ϕ|K is an isomorphism, ker(ϕ) is discrete. Moreover, as ϕ(Gp,q) contains both (1,0,0) and p×q, it follows that ϕ is surjective.

Let β be the inner automorphism of G given by (1,0,0). Then ϕα=βϕ and β is expansive. Since the kernel of ϕ is discrete, expansiveness of α follows from Theorem A. As the open subgroup Kp×q satisfies an ascending chain condition on closed subgroups (see, e.g., [6, Proposition 3.2]), Uα is closed by [26, Lemma 3.2].

In case UαUα-1 is a group, we will now show that ϕ is an isomorphism which would lead to a contradiction as Gp,q is not solvable but G is solvable.4 Suppose N:=UαUα-1 is a group. Then ϕ|N is an isomorphism of N with p×q (using the fact that Uαq and Uα-1p).5 Now the group generated by t and N is an open subgroup of G containing both t and a, hence Gp,q=t,N=tN (as t normalizes N). This implies that ϕ is an isomorphism. ∎

Acknowledgements

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.

References

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    Baumgartner U. and Willis G. A., Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248.

    • Crossref
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    Borel A., Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.

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    Bourbaki N., Lie Groups and Lie Algebras. Chapters 1–3, Springer, Berlin, 1989.

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    Elder M. and Willis G. A., Totally disconnected groups from Baumslag–Solitar groups, preprint 2013, http://arxiv.org/abs/1301.4775.

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    Glöckner H., Contraction groups for tidy automorphisms of totally disconnected groups, Glasg. Math. J. 47 (2005), 329–333.

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    Glöckner H., Locally compact groups built up from p-adic Lie groups, for p in a given set of primes, J. Group Theory 9 (2006), 427–454.

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    Glöckner H., Invariant manifolds for analytic dynamical systems over ultrametric fields, Expo. Math. 31 (2013), 116–150.

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    Glöckner H., Invariant manifolds for finite-dimensional non-archimedean dynamical systems, Advances in Non-Archimedean Analysis, Contemp. Math. 665, American Mathematical Society, Providence (2016), 73–90.

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    Glöckner H. and Willis G. A., Topologization of Hecke pairs and Hecke C*-algebras, Topol. Proc. 26 (2001–2002), 565–591.

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    Glöckner H. and Willis G. A., Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math. 643 (2010), 141–169.

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    Hewitt E. and Ross K. A., Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, New York, 1979.

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    Jacobson N., A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), 281–283.

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    Jaworski W., On contraction groups of automorphisms of totally disconnected locally compact groups, Israel J. Math. 172 (2009), 1–8.

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    Kitchens B. P., Expansive dynamics on zero-dimensional groups, Ergodic Theory Dynam. Systems 7 (1987), 249–261.

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    Lam P., On expansive transformation groups, Trans. Amer. Math. Soc. 150 (1970), 131–138.

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    Mal’cev A. I., On the faithful representation of infinite groups by matrices., Transl. Amer. Math. Soc. (2), 45 (1965), 1–18.

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    Margulis G. A., Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin, 1991.

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    Meskin S., Nonresidually finite one-relator groups, Trans. Amer. Math. Soc. 164 (1972), 105–114.

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    Ratner M., Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups, Duke Math. J. 77 (1995), 275–382.

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    Schikhof W. H., Ultrametric Calculus, Cambridge University Press, Cambridge, 1984.

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    Schlichting G., On the periodicity of group operations, Group Theory (Singapore 1987), De Gruyter, Berlin (1989), 507–517.

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    Schmidt K., Dynamical Systems of Algebraic Origin, Progr. Math. 128, Birkhäuser, Basel, 1995.

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    Serre J.-P., Lie Algebras and Lie Groups, Springer, Berlin, 1992.

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    Siebert E., Contractive automorphisms on locally compact groups, Math. Z. 191 (1986), 73–90.

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    Siebert E., Semistable convolution semigroups and the topology of contraction groups, Probability Measures on Groups. IX (Oberwolfach 1988), Lecture Notes in Math. 1379, Springer, Berlin (1989), 325–343.

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    Wang J. S. P., The Mautner phenomenon for p-adic Lie groups, Math. Z. 185 (1984), 403–412.

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    Weil A., Basic Number Theory, 2nd ed., Springer, Berlin, 1973.

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    Wesolek P., Elementary totally disconnected locally compact groups, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1387–1434.

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    Willis G. A., The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341–363.

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    Willis G. A., Further properties of the scale function on a totally disconnected group, J. Algebra 237 (2001), 142–164.

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    Willis G. A., The nub of an automorphism of a totally disconnected, locally compact group, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1365–1394.

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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 βAut(G)β(UαUα-1).

3

Let Θ be the set of continuous homomorphisms from p to Uα or Uα-1. By G=UαUα-1 and Lemma 7.3 (a), the first hypothesis of Lemma 7.4 is satisfied. Since L(G)=L(Uα)+L(Uα-1) and L(Uα)L(Uα-1)={θ(0):θΘ} by Lemma 7.3 (a) and “(a)  (c)” in Lemma 7.1, also the second hypothesis of Lemma 7.4 is satisfied.

4

Alternatively, BS(p,q) would be a finitely generated linear group then and hence residually finite by [16]. But BS(p,q) is not residually finite for primes pq, see [18].

5

In fact, V=K is tidy for α with V-q, V+p and V0={1} (see [4]), whence we have Uα=V--q and Uα-1=V++p (cf. Lemma 1.1 (b)).

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  • [1]

    Baumgartner U. and Willis G. A., Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248.

    • Crossref
    • Export Citation
  • [2]

    Borel A., Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.

  • [3]

    Bourbaki N., Lie Groups and Lie Algebras. Chapters 1–3, Springer, Berlin, 1989.

  • [4]

    Elder M. and Willis G. A., Totally disconnected groups from Baumslag–Solitar groups, preprint 2013, http://arxiv.org/abs/1301.4775.

  • [5]

    Glöckner H., Contraction groups for tidy automorphisms of totally disconnected groups, Glasg. Math. J. 47 (2005), 329–333.

    • Crossref
    • Export Citation
  • [6]

    Glöckner H., Locally compact groups built up from p-adic Lie groups, for p in a given set of primes, J. Group Theory 9 (2006), 427–454.

  • [7]

    Glöckner H., Invariant manifolds for analytic dynamical systems over ultrametric fields, Expo. Math. 31 (2013), 116–150.

    • Crossref
    • Export Citation
  • [8]

    Glöckner H., Invariant manifolds for finite-dimensional non-archimedean dynamical systems, Advances in Non-Archimedean Analysis, Contemp. Math. 665, American Mathematical Society, Providence (2016), 73–90.

  • [9]

    Glöckner H. and Willis G. A., Topologization of Hecke pairs and Hecke C*-algebras, Topol. Proc. 26 (2001–2002), 565–591.

  • [10]

    Glöckner H. and Willis G. A., Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math. 643 (2010), 141–169.

  • [11]

    Hewitt E. and Ross K. A., Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, New York, 1979.

  • [12]

    Jacobson N., A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), 281–283.

    • Crossref
    • Export Citation
  • [13]

    Jaworski W., On contraction groups of automorphisms of totally disconnected locally compact groups, Israel J. Math. 172 (2009), 1–8.

    • Crossref
    • Export Citation
  • [14]

    Kitchens B. P., Expansive dynamics on zero-dimensional groups, Ergodic Theory Dynam. Systems 7 (1987), 249–261.

    • Crossref
    • Export Citation
  • [15]

    Lam P., On expansive transformation groups, Trans. Amer. Math. Soc. 150 (1970), 131–138.

    • Crossref
    • Export Citation
  • [16]

    Mal’cev A. I., On the faithful representation of infinite groups by matrices., Transl. Amer. Math. Soc. (2), 45 (1965), 1–18.

    • Crossref
    • Export Citation
  • [17]

    Margulis G. A., Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin, 1991.

  • [18]

    Meskin S., Nonresidually finite one-relator groups, Trans. Amer. Math. Soc. 164 (1972), 105–114.

    • Crossref
    • Export Citation
  • [19]

    Ratner M., Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups, Duke Math. J. 77 (1995), 275–382.

    • Crossref
    • Export Citation
  • [20]

    Schikhof W. H., Ultrametric Calculus, Cambridge University Press, Cambridge, 1984.

  • [21]

    Schlichting G., On the periodicity of group operations, Group Theory (Singapore 1987), De Gruyter, Berlin (1989), 507–517.

  • [22]

    Schmidt K., Dynamical Systems of Algebraic Origin, Progr. Math. 128, Birkhäuser, Basel, 1995.

  • [23]

    Serre J.-P., Lie Algebras and Lie Groups, Springer, Berlin, 1992.

  • [24]

    Siebert E., Contractive automorphisms on locally compact groups, Math. Z. 191 (1986), 73–90.

    • Crossref
    • Export Citation
  • [25]

    Siebert E., Semistable convolution semigroups and the topology of contraction groups, Probability Measures on Groups. IX (Oberwolfach 1988), Lecture Notes in Math. 1379, Springer, Berlin (1989), 325–343.

  • [26]

    Wang J. S. P., The Mautner phenomenon for p-adic Lie groups, Math. Z. 185 (1984), 403–412.

    • Crossref
    • Export Citation
  • [27]

    Weil A., Basic Number Theory, 2nd ed., Springer, Berlin, 1973.

  • [28]

    Wesolek P., Elementary totally disconnected locally compact groups, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1387–1434.

    • Crossref
    • Export Citation
  • [29]

    Willis G. A., The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341–363.

    • Crossref
    • Export Citation
  • [30]

    Willis G. A., Further properties of the scale function on a totally disconnected group, J. Algebra 237 (2001), 142–164.

    • Crossref
    • Export Citation
  • [31]

    Willis G. A., The nub of an automorphism of a totally disconnected, locally compact group, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1365–1394.

    • Crossref
    • Export Citation
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