aa r X i v : . [ m a t h . D S ] O c t Orbits of Distal actions on Locally CompactGroups
Riddhi ShahOctober 22, 2010
Abstract
We discuss properties of orbits of (semi)group actions on locallycompact groups G , In particular, we show that if a compactly gener-ated locally compact abelian group acts distally on G then the closureof each of its orbits is a minimal closed invariant set (i.e. the action has[MOC]). We also show that for such an action distality is preserved ifwe go modulo any closed normal invariant subgroup and hence [MOC]is also preserved. We also show that any semigroup action on G has[MOC] if and only if the corresponding actions on a compact invariantmetrizable subgroup K and on the quotient space G/K has [MOC].
Let X be a Hausdorff space and Γ be a (topological) semigroup acting contin-uously on X by continuous self-maps. The action of Γ on X is said to be distal if for any two distinct points x, y ∈ X , the closure of { ( γ ( x ) , γ ( y )) | γ ∈ Γ } does not intersect the diagonal { ( a, a ) | a ∈ X } ; it is said to be pointwisedistal if for each γ ∈ Γ, the action of { γ n } n ∈ N on X is distal. The Γ-actionon X is said to have [MOC] (minimal orbit closures) if the closure of everyΓ-orbit is a minimal closed Γ-invariant set, i.e. for x, y ∈ X , if y ∈ Γ( x ) thenΓ( y ) = Γ( x ). The notion of distality was introduced by Hilbert (cf. Ellis[5], Moore [11]) and studied by many in different contexts, (see Abels [1]-[2],Furstenberg [6], Raja-Shah [15] and the references cited therein).Let G be a locally compact (Hausdorff) group and let e denote the identityof G . Let Γ be a semigroup acting continuously on G by endomorphisms.1hen Γ-action on G is distal if and only if e Γ x for all x ∈ G \ { e } .Note that if Γ-action on G has [MOC], then it is distal; for if e ∈ Γ x , then { e } = Γ e = Γ x and hence x = e . What we are interested in is the converse.If Γ-action on G is distal, does it have [MOC]? The answer is known to beaffirmative in any of the following cases: (1) G is compact (2) Γ is compact,(3) G is a connected Lie group and Γ is a subgroup of Aut( G ) (4) G isdiscrete, or more generally, all Γ-orbits are closed. If Γ is a group and if Γ ′ isa closed co-compact normal subgroup, then Γ-action on G has [MOC] if andonly if Γ ′ -action on G has [MOC] (cf. [11]); it is easy to see that the sameequivalence is true for distality. For a locally compact group G and a groupΓ ⊂ Aut( G ) acting distally on G , the answer to the above question is notknown. But in case of a certain kind of Γ, we get the following: Theorem 1.1
Let G be a locally compact group and let Γ be a compactlygenerated locally compact abelian group such that Γ acts on G by automor-phisms. Then the following are equivalent:1. The Γ -action on G is distal2. The Γ -action on G has [MOC] . Let us now discuss general actions on compact spaces. For a compactspace K , let Γ be a semigroup of continuous bijective self-maps of K . ThenΓ is a subsemigroup of C ( K ), the group of all continuous bijective self-mapson K . Let [Γ] be the group generated by Γ in C ( K ). We know that Γ-acts distally on K if and only if E (Γ), the closure of Γ in K K with weaktopology, is a group (cf. [5]); it is obviously compact since K K is so. Then E (Γ) = E ([Γ]). Moreover, for any x ∈ K , Γ( x ) = E (Γ)( x ) = E ([Γ])( x ). Sofor a compact space K and Γ and [Γ] as above, the following are equivalent:1. Γ-action on K is distal.2. [Γ]-action on K is distal.3. Γ-action on K has [MOC].4. [Γ]-action on K has [MOC].In particular, if G is a locally compact group and Γ a semigroup in Aut( G )such that Γ keeps a closed co-compact subgroup H of G invariant (i.e. γ ( H ) = H for all γ ∈ Γ), then the above equivalence is also true for the actions of Γ2nd [Γ] on
G/H . Note that for any Γ-action on G , the coresponding Γ-actionon the homogeneous space G/H = { xH | x ∈ G } is canonically defined as γ ( xH ) = γ ( x ) H for all γ ∈ Γ; it is well-defined since H is Γ-invariant.In [15], it is shown that distality of a semigroup action is preserved byfactor actions modulo compact invariant subgroups. We show that a similarresult holds for [MOC], (see also Remark 2.2). Theorem 1.2
Let G be a locally compact group and let Γ be a subsemigroupof Aut( G ) . Let K be a compact metrizable Γ -invariant subgroup of G . Then Γ -action on G has [MOC] if and only if Γ -action on both K and G/K has [MOC] . The following result is about factor actions modulo closed normal invari-ant subgroups.
Theorem 1.3
Let G and Γ be as in Theorem 1.1. Let H be a closed normal Γ -invariant subgroup of G . Then Γ -action on G has [MOC] if and only if Γ -action on both H and G/H has [MOC] . We will later show that a similar result holds for distality for a largerclass of Γ.A locally compact group G is said to be distal (resp. pointwise distal ) ifthe conjugacy action of G on G is distal (resp. pointwise distal). A distalgroup is obviously pointwise distal. It can easily be seen that the class ofdistal groups is closed under compact extensions. Abelian groups, discretegroups and compact groups are obviously distal. Nilpotent groups, connectedgroups of polynomial growth are distal (cf. [17]) and p-adic Lie groups of type R and p-adic Lie groups of polynomial growth are pointwise distal (cf. Raja[12] and [13]).In [15], we have shown that any locally compact group is pointwise distalif and only if it has shifted convolution property; i.e. for any probabilitymeasure µ on G , whose concentration functions do not converge to zero,there exists x ∈ supp µ , the support of µ , such that µ n x − n → ω H , the Haarmeasure of some compact group H which is normalised by supp µ . For aprobability measure µ on G , the n -th convolution function of µ is defined as f n ( µ, C ) = sup g ∈ G µ n ( Cg ), for any compact subset C of G . We say that theconcentration functions of µ do not converge to zero if there exists a compactset C such that f n ( µ, C ) n → ∞ , (see [15] for more details). Thefollowing corollary is a consequence of Theorem 6.1 of [15] and Theorem 1.1.3 orollary 1.4 Let G be a locally compact group. Then the following areequivalent:1. G is pointwise distal.2. G has shifted convolution property.3. For every g ∈ G , the conjugation action of { g n } n ∈ Z on G has [MOC] . A locally compact group G is said to be a generalised F C − - group (resp. F C − - nilpotent ) if G has closed normal subgroups { G = G , . . . , G n = { e }} such that G i +1 ⊂ G i and G i /G i +1 is a compactly generated group with rela-tively compact conjugacy classes (resp. every orbit of the conjugacy action of G on G i /G i +1 is relatively compact) for all i = 0 , , . . . , n −
1. Any compactlygenerated group G has polynomial growth if and only if it is F C − -nilpotent;and it is a generalised F C − -group (cf. [10]). Any compactly generated abeliangroup (resp. any polycyclic group) is a generalised F C − -group. More gener-ally, any compactly generated group with polynomial growth is a generalised F C − -group. Note that generalised F C − -groups are compactly generated (cf.[10], Proposition 2).Recall that a subgroup Γ of Aut( G ) is said to be equicontinuous (at e ) ifand only if there exists a neighbourhood base at e consisting of Γ-invariantneighbourhoods; in case of totally disconnected groups, this is equivalentto the existence of a neighbourhood base at e consisting of compact openΓ-invariant subgroups. If Γ is compact, then it is easy to see that Γ isequicontinuous. If G is a totally disconnected group and if Γ has a polycyclicsubgroup of finite index and it acts distally on G , then Γ is equicontinuous(cf. [9], Corollary 2.4). If any group Γ acts on G by automorphisms andits image in Aut( G ) is equicontinuous then we say that Γ-action on G isequicontinuous.For a totally disconnected locally compact group G , we have the followingresult: Proposition 1.5
Let G be a totally disconnected locally compact group andlet Γ be a generalised F C − -group which acts on G by automorphisms. Thenthe following are equivalent.1. Γ -action on G is distal.2. Γ -action on G has [MOC] . . Γ -action on G is equicontinuous. In Section 2, we discuss factor actions modulo compact (resp. closed nor-mal) invariant groups and prove Theorem 1.2, Proposition 1.5 and an ana-logue of Theorem 1.3 for distal actions of a more general class of groups. InSection 3, we prove the equivalence of distality and [MOC] of certain actions,namely, Theorem 1.1. Note that if Γ acts on G by automorphisms, for con-venience, Γ is often equated with its image in Aut( G ), whenever there is noloss of any generality. In this section we discuss [MOC] of factor actions modulo compact invari-ant groups and modulo closed normal invariant groups. We first show that[MOC] is preserved if we go modulo a compact invariant subgroup by provingTheorem 1.2. Before that we prove a proposition which proves a special caseof the theorem in case the compact subgroup is a Lie group.
Proposition 2.1
Let G be a locally compact group and let Γ be a subsemi-group of Aut( G ) . Let K and L be compact Γ -invariant subgroups of G suchthat L is a normal subgroup of K and K/L is a Lie group. Then Γ -action on G/L has [MOC] if and only if Γ -action on both G/K and
K/L has [MOC] . Proof Step 1
Let G , Γ, K and L be as in the hypothesis. One wayimplication “only if” is easy to prove. Suppose Γ-action on G/L has [MOC].Then clearly Γ-action on
K/L also has [MOC], as K is closed and Γ-invariant.Now we want to show that Γ-action on G/K has [MOC]. Let x ∈ G and let yK ∈ Γ( xK ) in G/K for some y ∈ G . Then yK ⊂ Γ( x ) K = Γ( x ) K andhence yk ∈ Γ( x ) for some k ∈ K . In particular, we get that ykL ⊂ Γ( x ) L =Γ( x ) L as L is compact. Hence ykL ∈ Γ( xL ) in G/L . Since Γ-action on
G/L has [MOC], we get that Γ( xL ) = Γ( ykL ) and hence x ∈ Γ( y ) K as k ∈ K , L ⊂ K and both L and K are Γ-invariant. This implies that xK ∈ Γ( yK ) in G/K and hence Γ-action on
G/K has [MOC]. Note that the condition that
K/L is a Lie group is not used in the proof of the “only if” statement.
Step 2
Now we prove the “if” statement. Suppose Γ-action on both
G/K and
K/L has [MOC]. This implies that Γ-action on both
G/K and
K/L isdistal and hence Γ-action on
G/L is distal; (this is easy to see from the proofof Theorem 3.1 in [15]). 5or any g ∈ G , let g ′ = gL . The map g g ′ is a continuous proper mapfrom G to G/L . Let x ∈ G and let y ′ ∈ Γ( x ′ ) for some y ∈ G . We wantto show that x ′ ∈ Γ( y ′ ). Then yK ∈ Γ( xK ), and as Γ-action on G/K has[MOC], xK ∈ Γ( yK ). This implies that xk ∈ Γ( y ) for some k ∈ K , andhence, x ′ k ′ ∈ Γ( y ′ ). Let { γ d } and { β d } be nets in Γ such that γ d ( x ′ ) → y ′ and β d ( y ′ ) → x ′ k ′ . Step 3
Let Γ be the closure of image of Γ in Aut( K/L ). Suppose Γ iscompact. Then Γ , being a compact semigroup, is a group. Let β and γ belimit points of images of { β d } and { γ d } in Γ respectively. Then γ d ( x ′ k ′ ) → y ′ γ ( k ′ ) ∈ Γ( y ′ ) and β d ( y ′ γ ( k ′ )) → x ′ k ′ α ( k ′ ) ∈ Γ( y ′ ) , where α = βγ ∈ Aut(
G/K ). Similarly we get that for k n = k ′ α ( k ′ ) · · · α n − ( k ′ ) ∈ K/L, x n = x ′ k n ∈ Γ( y ′ ) , for all n ∈ N . As Γ is a compact group, there exists a sequence { n j } ⊂ N such that α n j → I , the identity of Aut( K/L ). Passing to a subsequence if necessary,we may assume that k n j → c ′ = cL ∈ K/L , for some c ∈ K . Hence x ′ c ′ ∈ Γ( y ′ ). Now as α n j → I , k n j = k n j α n j ( k n j ) → ( cL ) = c L. Similarly, for all m ∈ N , k mn j = k n j α n j ( k n j ) · · · α ( m − n j ( k n j ) → c m L ∈ K/L and xc m L ∈ Γ( yL ). Since K/L is a compact (Lie) group, e ′ = eL is in theclosure of { c m L } m ∈ N in K/L and hence x ′ ∈ Γ( y ′ ), i.e. Γ( x ′ ) = Γ( y ′ ). HenceΓ-action on G/L has [MOC].In particular, since K L/L is the connected component of
K/L , K/K L is finite, and hence, Aut( K/K L ) is finite. Arguing as above for K L in placeof L , we get that G/K L has [MOC] and we may assume that K = K L ,i.e. K/L is connected.
Step 4
Now Let Z be the subgroup of K such that L ⊂ Z and Z/L is thecenter of
K/L . Then Z and Z L are closed and Γ-invariant. Moreover, K/Z is a connected semisimple Lie group and hence its automorohism group iscompact. Therefore arguing as in Step 3 for Z in place of L , we get that6-action on G/Z has [MOC], and since
Z/Z L is finite, Γ-action on G/Z L also has [MOC]. Now replacing K by Z L , we may assume that K/L is aconnected abelian Lie group.Let [Γ] be the group generated by Γ in Aut(
K/L ). Then [Γ] also actsdistally on
K/L . By Lemma 2.5 of [2], there exists a finite set of compact(normal) [Γ]-invariant subgroups { K , . . . , K n } in K such that K = K ⊃ K ⊃ · · · ⊃ K n = L and the image of [Γ] in Aut( K i /K i +1 ) is finite for each i ∈ { , . . . , n − } . Arguing as in Step 3 for K in place of L , we get that Γ-action on G/K , has [MOC]. Since the image of Γ in Aut( K i /K i +1 ) is finite,using the above argument repeatedly for K i /K i +1 in place of K/L , we getthat Γ-action on
G/K i +1 has [MOC], 1 ≤ i ≤ n −
1. Since K n = L , we havethat Γ-action on G/L has [MOC]. (cid:3)
Proof of Theorem 1.2
Let G , Γ and K be as in the hypothesis. As in theproof of Proposition 2.1, the “only if” statement is obvious. Now we provethe “if” statement. Suppose that Γ-action on G/K and K has [MOC]. HenceΓ-actions on G/K , K and G are distal. Let K consists of closed (compact) Γ-invariant subgroups C of K such that Γ-action on G/C has [MOC]. Then K is nonempty as K belongs to K . We put an order on K by set inclusion. Let A = { K d } be a totally ordered subset of K . We show that K ′ = ∩ K d ∈ K .For any x ∈ G and y ∈ Γ( x ) K ′ , we show that Γ( x ) K ′ = Γ( y ) K ′ . We knowthat Γ( x ) K d = Γ( y ) K d for each d . First we show that ∩ d Γ( x ) K d = Γ( x ) K ′ .One way inclusion is obvious. Let a ∈ ∩ d Γ( x ) K d . Then C d = Γ( x ) ∩ aK d = ∅ for all d . Here, A ′ = { C d } is a collection of compact sets and intersectionof finitely many subsets in A ′ is nonempty since A is totally ordered. Hence ∩ d C d is nonempty. But ∩ d C d = ∩ d (Γ( x ) ∩ aK d ) = Γ( x ) ∩ ( ∩ d aK d ) = Γ( x ) ∩ aK ′ = ∅ . Hence a ∈ Γ( x ) K ′ . Therefore, ∩ d Γ( x ) K d = Γ( x ) K ′ . Similarly, ∩ d Γ( y ) K d =Γ( y ) K ′ . This implies that Γ( x ) K ′ = Γ( y ) K ′ and hence Γ-action on G/K ′ has [MOC], i.e. K ′ ∈ K .By Zorn’s Lemma, there exists a minimal element in K , say M . Here, M is a compact Γ-invariant subgroup of K such that Γ-action on G/M has [MOC] and there is no proper subgroup of M in K . We show that M = { e } . If possible suppose M is nontrivial. Since M ⊂ K is compactand metrizable and since Γ-action on M is distal, it is not ergodic and thereexists a (nontrivial) irreducible unitary representation χ of M such that χ Γis finite upto equivalence classes (cf. [3], Theorem 2.1, see also [14] as the7ction of the group [Γ] generated by Γ is also distal). Let L = ∩ γ ∈ Γ ker( χγ )Then L is a proper closed (compact) normal Γ-invariant subgroup of M andsince χ Γ is finite upto equivalence classes,
M/L is a (compact) Lie group.Moreover, Γ-action on
M/L is distal (cf. [15], Theorem 3.1) and hence it has[MOC]. By Proposition 2.1, we get that Γ-action on
G/L has [MOC]. Hence L ∈ K , a contradiction to the minimality of M in K . Hence M = { e } andΓ-action on G has [MOC]. (cid:3) Remark 2.2
1. In Theorem 1.2, if G is first countable then K is also firstcountable and hence metrizable.2. Theorem 1.2 holds in case Γ is a locally compact σ -compact group, (fore.g. Γ = Z ) and K is not (necessarily) metrizable. As in this case, the group M as above is not necessarily metrizable. Here, Γ ⋉ M is locally compact and σ -compact and hence M has arbitrarily small compact normal Γ-invariantsubgroups M d such that ∩ d M d = { e } and M/M d is second countable andhence metrizable (cf. [7], Theorem 8.7). Now from Theorem 3.1 of [15], ifΓ-action on M is distal then the corresponding Γ-action on M/M d is alsodistal and hence not ergodic and we get a proper closed normal Γ-invariantsubgroup (of M/M d , and hence,) of M , denote it by L again, such that M/L is a Lie group. Now the assertion is obvious from the above proof. Note thatany compactly generated locally compact group is σ -compact.The following corollary follows from Theorem 3.1 in [15], Theorem 1.1 in[2] and Theorem 1.2 above since every connected locally compact group hasa unique maximal compact normal (characteristic) subgroup such that thequotient is a connected Lie group. Corollary 2.3
Let G be a connected locally compact first countable group.Let Γ be a subgroup of Aut( G ) . Then Γ -action on G is distal if and only ifit has [MOC] . We now show that [MOC] is preserved by factors modulo closed normalinvariant group. Before that we prove Proposition 1.5 and a Lemma whichwill be useful in proving Theorem 2.5 below and also Theorem 1.1.
Proof of Proposition 1.5
Let G be a locally compact totally disconnectedgroup and let Γ be a generalised F C − -group acting on G by automorphisms.Let Γ = { γ ∈ Γ | γ ( x ) = x for all x ∈ G } . Then Γ is a closed normalsubgroup of Γ, Γ / Γ is isomorphic to a subgroup of Aut( G ). Also, Γ / Γ is a8eneralised F C − -group. It is easy to see that we can replace Γ by Γ / Γ andassume that Γ ⊂ Aut( G ). We prove that (1) ⇒ (3) ⇒ (2) ⇒ (1).Suppose Γ acts distally on G . As Γ is totally disconnected, it has acompact open normal subgroup C such that Γ /C has a polycyclic subgroupof finite index (cf. [10]). Since C is equicontinuous, By Lemma 2.3 of [9],Γ-action on G is also equicontinuous, (see also ‘Note added in Proof’ in [9]for non-metrizable groups). Now G has a neighbourhood base at e consistingof open compact subgroups K d which are Γ-invariant and ∩ d K d = { e } . Since G/K d is discrete, Γ-action on G/K d has [MOC]. Let x ∈ G and y ∈ Γ( x ).Then Γ( x ) K d = Γ( x ) K d = Γ( y ) K d = Γ( y ) K d as K d is open for all d . Γ( x ) = ∩ d Γ( x ) K d = ∩ d Γ( y ) K d = Γ( y ). This proves that Γ-action on G has [MOC].We know that [MOC] implies distality. (cid:3) Lemma 2.4
Let G be a locally compact group. Let Γ be a group acting on G by automorphisms. Suppose that Γ -action on G/G is equicontinuous.Then there exist open (resp. compact) Γ -invariant subgroups H d (resp. K d )such that H d = K d G , K d is the maximal compact normal subgroup of H d , K d ∩ G = ∩ d K d is the maximal compact normal Γ -invariant subgroup of G .In particular, if G has no nontrivial compact normal subgroup, then K d istotally disconnected and H d = K d × G , a direct product, for all d . Proof
Since Γ-action on
G/G is equicontinuous, there exist open almostconnected Γ-invariant subgroups H d such that { H d /G } form a neighbour-hood base at the identity in G/G consisting of compact open subgroups.Choose H = H d for some fixed d . Since H is Lie projective, there ex-ists a compact normal subgroup C in H such that H/C is a Lie group withfinitely many connected components. Hence H has a maximal compact nor-mal subgroup, we denote it by C again. Then C is characteristic in H , andin particular, it is Γ-invariant. Let H ′ = CG . It is an open Γ-invariantsubgroup in G and K = C ∩ G is the maximal compact normal subgroupof G . Since H ′ /G is compact and open in G/G , passing to a subnet, wemay assume that H d ⊂ H ′ for all d . Let K d = C ∩ H d . Then K d is a com-pact normal Γ-invariant subgroup in H d and H d = K d G as G ⊂ H d . As K = C ∩ G ⊂ H d , K = K d ∩ G and K d is the maximal compact normalsubgroup in H d for every d . Also, since ∩ d H d = G , we get that ∩ d K d = K .Moreover, if K d ∩ G = K is trivial, then K d is totally disconnected and H d = K d × G as both K d and G are normal in H d , for all d . (cid:3)
9o prove Theorem 1.3, in view of Theorem 1.1, it is enough if we provethe same statement for distal actions. Here, we prove the following for distalactions of a more general class of groups.
Theorem 2.5
Let G be a locally compact group and Γ be a generalised F C − -group which acts on G by automorphsm. Let H be a closed normal Γ -invariant subgroup. Then Γ -action on G is distal if and only if Γ -actionson both H and G/H are distal.
Proof
Let G , H and Γ be as in the hypothesis. Suppose Γ-actions on G/H and H are distal. Then it is easy to see that Γ-action on G is distal.Now we prove the converse. Suppose Γ-action on G is distal and henceΓ-action on H is also distal. As in the proof of Theorem 1.5, we may assumethat Γ ⊂ Aut( G ). We prove that Γ-action on G/H is distal. By Theorem 3.3of [15], Γ action on
G/G is distal and hence equicontinuous (by Proposition1.5). There exists an open Γ invariant subgroup L in G such that L = KG ,where K is the maximal compact normal Γ-invariant subgroup of L (cf.Lemma 2.4). We know that G/L is discrete, and hence, so is
G/HL , where HL is an open Γ-invariant subgroup. Therefore, it is enough to prove thatΓ acts distally on HL/H . Since
HL/H is isomorphic to L/ ( L ∩ H ), withoutloss of any generality, we may assume that G = L = KG and K is themaximal compact normal Γ-invariant subgroup in G . In particular, G/K isa connected Lie group without any nontrivial compact normal subgroup.Here, HK and K ∩ H are closed, normal and Γ-invariant subgroups. ByTheorem 3.1 of [15] we know that Γ-action is distal on G/K , HK/K and on K/ ( K ∩ H ); the latter is isomorphic to HK/H . Hence it is enough to provethat Γ-action is distal on
G/HK which is isomorphic to (
G/K ) / ( HK/K ).Replacing G by G/K and H by HK/K , we may assume that G is aconnected Lie group and H is a closed normal (Lie) subgroup and G , andhence H , has no nontrivial compact normal subgroup. Let G be the Liealgebra of G . Since Γ-action on G is distal, so is the corresponding actionof { d γ | γ ∈ Γ } on G (cf. [2], Theorem 1.1). Equivalently, the eigenvaluesof d γ are of absolute value 1, for all γ ∈ Γ (cf. [1], Theorem 1 ′ ). Since H isnormal and Γ-invariant, the Lie algebra H of H is a Lie subalgebra whichis an ideal invariant under d γ , for all γ ∈ Γ, and the Lie algebra of
G/H isisomorphic to G / H . Then the eigenvalues of d γ on G / H are also of absolutevalue 1 for all γ ∈ Γ. Hence Γ-acts distally on
G/H (cf. [1], [2]). (cid:3) Distality and [MOC]
In this section we show that distality and [MOC] of Γ-action on any locallycompact group are equivalent if Γ is a locally compact, compactly generatedabelian (resp. Moore) group acting on the group by automorphisms. We firstprove a proposition which will be useful in proving Theorem 1.1.
Proposition 3.1
Let G and Γ be as in Theorem 1.1. Suppose that the Γ -action on G is distal. Given a net { γ d } in Γ , let M = { g ∈ G | { γ d ( g ) } d is relatively compact } . Then M is a closed Γ -invariant subgroup. Proof
It is obvious that M is a subgroup and it is Γ-invariant since Γ isabelian. Therefore M is also a Γ-invariant subgroup. If M is trivial, then M = M . Suppose M is a nontrivial subgroup of G . Without loss of anygenerality, we may assume that G = M , i.e. M is dense in G . Step 1
By Theorem 3.3 of [15], Γ-action on
G/G is distal. Since Γ isa compactly generated locally compact abelian group, it is a generalised F C − -group. By Proposition 1.5, Γ-action on G/G has [MOC] and Γ-actionon G/G is equicontinuous. By Lemma 2.4, there exists an open (resp.compact) Γ-invariant subgroups H (resp. K ) such that H = KG , where K is the maximal compact normal subgroup of H . Since H is open and Γ-invariant, it is enough to show that H ⊂ M and hence, we may assume that G = H . Here, since K is a maximal compact normal Γ-invariant subgroup, K ⊂ M and G/K is a connected Lie group without any nontrivial compactsubgroup. Moreover, Γ action on
G/K is distal (cf. [15], Theorem 3.1). Let π : G → G/K be the natural projection. Since K is compact, π ( M ) = { gK ∈ G/K | { γ d ( gK ) } d is relatively compact in G/K } and M is closed ifand only if π ( M ) is closed. Moreover, π ( M ) is dense in G/K . Now, we mayreplace G by G/K and assume that G is a connected Lie group without anynontrivial compact normal subgroup and Γ ⊂ Aut( G ). Step 2
Since G has no nontrivial compact central subgroup, Aut( G ) is almostalgebraic (as a subgroup of GL ( G )) (cf. [4]), where G is the Lie algebra of G . Let Γ ′ be the smallest almost algebraic subgroup containing Γ in Aut( G ).Here Γ ′ is a an open subgroup of finite index in the Zariski closure ˜Γ of Γ in GL ( G ), hence Γ ′ and ˜Γ have the same connected component of the identity.11t follows from Corollary 2.5 of [1], that the unipotent radical U of ˜Γ is aclosed co-compact normal subgroup of Γ ′ and as in the proof of Theorem 1.1in [2]. we have that U , and hence Γ ′ , has closed orbits in G . Step 3
We now prove that { γ d } d is relatively compact in Aut( G ). Suppose { γ d } d is not relatively compact in Aut( G ). Since Aut( G ) is a Lie group, thereexists a divergent sequence { γ ′ n } in the set { γ d } . We know that { γ ′ n ( g ) } isrelatively compact for all g in a dense subgroup M . There exists a countablesubgroup M ⊂ M which is dense in G . Passing to a subsequence if necessary,we may assume that { γ ′ n ( g ) } converges for all g ∈ M .Since G is a Lie group without any compact central subgroup of positivedimension, from Step 2, for every g ∈ M , there exists γ g ∈ Γ ′ such that { γ ′ n ( g ) } converges to γ g ( g ). Then γ − g γ ′ n ( g ) → g for every g ∈ M . Let V (resp. W ) be open relatively compact neighbourhoods of the identity e in G (resp. zero in G ) such that the exponential map from W to V is adiffeomorphism with log as its inverse. Let U be an open neighbourhoodof e in G such that U ⊂ V . Then ( d γ − g d γ ′ n )(log g ) → log g , and henced γ ′ n (log g ) → d γ g (log g ) for all g ∈ U ∩ M .In particular, { d γ ′ n ( w ) } converges for all w in a dense subset of log U ⊂ W in G . Since G is a vector space and log U is open, we get that any dense subsetof log U generates G and hence { d γ ′ n } converges in GL ( G ). Let γ ′ be the limitpoint of { d γ ′ n } in GL ( G ); it is a Lie algebra automorphism. Hence γ ′ = d γ for some γ ∈ Aut( G ). Then γ ′ n → γ . This is a contradiction to the aboveassumption that { γ ′ n } is divergent. Hence we have that { γ d } is relativelycompact in Aut( G ). This implies that { γ d ( x ) } d is relatively compact for all x ∈ G and G = M , i.e. M is closed. (cid:3) Remark 3.2
From the above proof it is clear that if G is a connected Liegroup without any nontrivial compact central subgroup and if Γ is a subgroupof Aut( G ) acting distally on G and if { γ d } ⊂ Γ is such that { γ d ( g ) } d isrelatively compact for all g in a dense subgroup of G , then { γ d } is relativelycompact in Aut( G ). Proof of Theorem 1.1
Let G be a locally compact group and let Γ be acompactly generated locally compact abelian group. Suppose Γ-action on G has [MOC], then we know that Γ-action on G is distal.Now suppose that the Γ-action on G is distal. We show that it has [MOC].Let x ∈ G and let y ∈ Γ( x ). We need to show that x ∈ Γ( y ). We have that12 d ( x ) → y for some { γ d } ⊂ Γ. Let M = { g ∈ G | { γ d ( g ) } d is relatively compact } . By Proposition 3.1, M is a closed Γ-invariant subgroup and x , and hence, y belongs to M . Without loss of any generality we may assume that M = G .In view of Theorem 1.2 and Remark 2.2, we can go modulo the maximalcompact normal subgroup of G which is characteristic in G and assumethat G is a Lie group without any nontrivial compact normal subgroup.Note that Γ is a generalised F C − -group and Γ-action on G/G is distal (byTheorem 3.3 of [15]). Hence from Proposition 1.5, we get that the action of Γon G/G is equicontinuous. Let H d = K d × G be open Γ-invariant subgroups,where K d are totally disconnected compact Γ-invariant subgroups such that ∩ d K d = { e } in G (cf. Lemma 2.4). Then passing to a subnet if necessary,we may assume that γ d ( x ) = yk d g d = yg d k d , where k d ∈ K d and g d ∈ G , k d → e , g d → e . In particular, we get that γ − d ( y ) = xγ − d ( k − d ) γ − d ( g − d ).We know that { γ d | G } is relatively compact, (see Remark 3.2). Let γ be alimit point of { γ d | G } in Aut( G ). Then γ − is a limit point of { γ − d | G } inAut( G ). Therefore, passing to a subnet if necessary, we get that γ − d ( g − d ) → γ − ( e ) = e and γ − d ( y ) = xk ′ d γ − d ( g − d ) → x where k ′ d = γ − d ( k − d ) ∈ K d and k ′ d → e as K d are Γ-invariant and ∩ d K d = { e } .In particular, x ∈ Γ( y ).Since the above is true for any x ∈ G and any y ∈ Γ( x ), closure of anyorbit is a minimal closed Γ-invariant set, i.e. Γ-action on G has [MOC]. (cid:3) A locally compact group G is said to be a central group or a Z -group if G/Z ( G ) is compact, where Z ( G ) is the center of G . It is said to be a Moore group if all its irreducible unitary representations are finite dimensional. Allabelain groups and all compact groups are Z groups and Z -groups are alsoMoore groups. A Moore group has normal subgroup H of finte index suchthat [ H, H ] is compact (cf. [16]). It is easy to see from this, that any Mooregroup G is F C − nilpotent as G = G , G = H , G = [ H, H ] and G = { e } .Since G /G is finite, and G /G is abelian and G /G is compact, we havethat the conjugacy action of G on G i /G i +1 has relatively compact orbits forall i = 0 , ,
2. Hence any compactly generated Moore group has polynomialgrowth and it is a generalised
F C − -group (cf. [10], Theorem 1, Lemma 1).13 orollary 3.3 Let G be a locally compact group and let Γ is a compactlygenerated Moore group acting on G by auotomorphisms. Then Γ -action on G is distal if and only if it has [MOC] . The proof of the above corollary is essentially the same as that of Theorem1.1. As Γ is a Moore group, it has a closed normal subgroup Γ of finiteindex whose commutator group is relatively compact. (cf. [16], Theorem 1).Then by Lemma 4.1 of [11], it is enough to show that Γ -action on G has[MOC]. Without loss of any generality, we may assume that [Γ , Γ] is relativelycompact and hence it is easy to see that the group M defined in the aboveproof is Γ-invariant. We will not repeat the proof here. Remark 3.4
1. From above, it is obvious that Theorem 1.1 holds for anycompactly generated locally compact group Γ such that its commutator sub-group is relatively compact. Moreover from Lemma 4.1 in [11] we know thatthe action of a group Γ on G has [MOC] if the action of any co-compact sub-group of Γ on G has [MOC]. Hence Theorems 1.1 and 1.3 hold for compactextensions of such a group Γ mentioned above, and in particular, for com-pact extensions of compactly generated abelian, or more generally, of Mooregroups.We conjecture that Theorem 1.1 holds for any generalised F C − -groups. Italready holds for the actions of such a group on totally disconnected groups,compact groups and connected groups. Acknowledgement
The author would like to thank H. Abels, S.G. Dani,Y. Guivarc’h and C.R.E. Raja for fruitful discussions. The author would alsolike to thank C.R.E. Raja for comments on a preliminary version which ledto improvement in the presentation of the manuscript.
References [1] H. Abels. Distal affine transformation groups. J. Reine Angew. Math. (1978), 294–300.[2] H. Abels, Distal automorphism groups of Lie groups. J. Reine Angew. Math. (1981), 82–87.[3] D. Berend, Ergodic semigroups of epimorphisms, Trans. Amer. Math. Soc. (1985), 393–407.
4] S. G. Dani, On automorphism groups of connected Lie groups. Manuscripta Math. (1992), 445–452.[5] R. Ellis, Distal transformation groups, Pacific J. Math. (1958), 401–405.[6] H. Furstenberg, The structure of distal flows. Amer. J. Math. (1963), 477–515.[7] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I. Structure of topologicalgroups, integration theory, group representations. Second edition. Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],115. Springer-Verlag, Berlin-New York, 1979.[8] K. H. Hofmann and S. A. Morris, The structure of compact groups. de GruyterStudies in Mathematics, 25. Walter de Gruyter & Co., Berlin, 1998.[9] W. Jaworksi and C. R. E. Raja, The Choquet-Deny theorem and distal propertiesof totally disconnected locally compact groups of polynomial growth, New York J.Math. (2007), 159-174.[10] V. Losert. On the structure of groups with polynomial growth II. J. London Math.Soc. (2) (2001), 640–654.[11] C. C. Moore. Distal affine transformation groups. Amer. J. of Math. (1968),733–751.[12] C. R. E. Raja, On classes of p -adic Lie groups. New York J. Math. (1999), 101–105.[13] C. R. E. Raja, On growth, recurrence and the Choquet-Deny theorem for p -adic Liegroups. Math. Z. 251 (2005), 827–847.[14] C. R. E. Raja, Distal actions and ergodic actions on compact groups. New York J.Math. (2009), 301–318.[15] C. R. E. Raja and Riddhi Shah. Distal actions and shifted convolution property.Israel Journal of Mathematics (2010), 391–412.[16] L. C. Robertson. A note on structure of Moore Groups. Bull. Amer. Math. Soc. (1969), 594–599.[17] J. Rosenblatt, A distal property of groups and the growth of connected locally com-pact groups, Mathematika (1979), 94–98.(1979), 94–98.