aa r X i v : . [ m a t h . G R ] M a r Actions of automorphism groups of Lie groups
S.G. DaniMarch 29, 2017
Introduction
The aim of this article is to discuss the actions on a connected Lie group by sub-groups of its automorphism group. The automorphism groups are themselves Liegroups (not necessarily connected) and the actions have, not surprisingly, playedan important role in the study of various topics, including geometry, dynamics,ergodic theory, probability theory on Lie groups, etc.We begin in § G ), G a Lie group, generalities about their subgroups, connected compo-nents, etc.. The automorphism group of a connected Lie group can be realisedas a linear group via association with the corresponding automorphism of the Liealgebra G of G , and § G ) to algebraic subgroups,and more generally “almost algebraic” subgroups, of GL( G ); in particular theconnected component of the identity in Aut( G ) turns out to be almost algebraicand this has found considerable use in the study of various topics discussed in thesubsequent sections. In § G ) on G , and especially conditions for them to be dense in G . In § G ) on G , and in § G ) play a role. The action of Aut( G ) on G induces ina natural way an action on the space of probability measures on G and propertiesof this action have played an important role in various questions in probabilitytheory on Lie groups, in terms of extending certain aspects of classical probabilitytheory to the Lie group setting. These are discussed in § ontents G ) . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Connected components of Aut( G ) . . . . . . . . . . . . . . . . . . 51.4 Locally isomorphic Lie groups . . . . . . . . . . . . . . . . . . . . 6 G ) . . . . . . . . . . . . . . . 82.3 Groups with Aut( G ) almost algebraic . . . . . . . . . . . . . . . . 92.4 Automorphisms preserving additional structure . . . . . . . . . . 102.5 Linearisation of the Aut( G ) action on G . . . . . . . . . . . . . . 102.6 Embedding of G in a projective space . . . . . . . . . . . . . . . . 112.7 Embedding in a vector space . . . . . . . . . . . . . . . . . . . . . 122.8 Algebraicity of stabilizers . . . . . . . . . . . . . . . . . . . . . . . 13 G )-actions with dense orbits . . . . . . . . . . . . . . . . . . 143.2 Connected subgroups of Aut( G ) with dense orbits . . . . . . . . . 153.3 Automorphisms with dense orbits . . . . . . . . . . . . . . . . . . 163.4 Z d -actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Discrete groups with dense orbits . . . . . . . . . . . . . . . . . . 183.6 Orbit structure of actions of some discrete groups . . . . . . . . . 193.7 Aut( G )-actions with few orbits . . . . . . . . . . . . . . . . . . . 20 Aut( G ) -actions 26 References 33
In this section we discuss various structural aspects of the automorphisms ofconnected Lie groups.
Let G be a connected Lie group and G be the Lie algebra of G . We denote byAut( G ) the group of all Lie automorphisms of G equipped with its usual topology,corresponding to uniform convergence on compact subsets (see [55] Ch. IX for somedetails). To each α ∈ Aut( G ) there corresponds a Lie automorphism dα of G , thederivative of α . We may view Aut( G ) in a natural way as a closed subgroup ofGL( G ), the group of nonsingular linear transformations of G , considered equippedwith its usual topology. Consider the map d : Aut( G ) → Aut( G ), given by α dα , for all α ∈ Aut( G ). As G is connected, dα uniquely determines α , andthus the map is injective.When G is simply connected the map is also surjective, by Ado’s theorem (cf.[53], Theorem 7.4.1). When G is not simply connected the map is in general notsurjective; in this case G has the form ˜ G/ Λ, where ˜ G is the universal coveringgroup of G and Λ is a discrete subgroup contained in the center of ˜ G , and theimage of Aut( G ) in Aut( G ) consists of those elements for which the correspondingautomorphism of Aut( ˜ G ) leaves the subgroup Λ invariant (note that G may beviewed also, canonically, as the Lie algebra of ˜ G ), and in particular it followsthat it is a closed subgroup. This implies in turn that d as above is a topologicalisomorphism (see also [55], Ch. IX, Theorem 1.2). We shall view Aut( G ) as asubgroup of Aut( G ), and in turn GL( G ), via the correspondence.3s a closed subgroup of GL( G ), Aut( G ) is Lie group, and in particular alocally compact group; it may be mentioned here that the question as to whenthe automorphism group of a general locally compact group is locally compact isanalysed in [90].We note that Aut( G ) is a real algebraic subgroup of GL( G ), namely the groupof R -points of an algebraic subgroup of GL( G ⊗ C ), defined over R . While Aut( G )is in general not a real algebraic subgroup (when viewed as a subgroup of GL( G ) asabove), various subgroups of Aut( G ) being algebraic subgroups plays an importantrole in various results discussed in the sequel. Aut( G ) We introduce here certain special classes of automorphisms which play an impor-tant role in the discussion in the following sections.Let G be a connected Lie group. For any subgroup H of G let Aut H ( G )denote the subgroup of Aut( G ) consisting of automorphisms leaving H invariant;viz. { τ ∈ Aut( G ) | τ ( H ) = H } . A subgroup H for which Aut H ( G ) = Aut( G ) iscalled a characteristic subgroup of G . Clearly, for any Lie group G the center of G , the commutator subgroup [ G, G ], the (solvable) radical, the nilradical are someof the characteristic subgroups of G . Similarly, together with any characteristicsubgroup, its closure, centraliser, normaliser etc. are characteristic subgroups.For each g ∈ G we get an automorphism σ g of G defined by σ g ( x ) = gxg − forall x ∈ G , called the inner automorphism corresponding to g . For a subgroup H of G we shall denote by Inn( H ) the subgroup of Aut( G ) defined by { σ h | h ∈ H } .For any Lie subgroup H , Inn( H ) is a Lie subgroup of Aut( G ); it is a normalsubgroup of Aut( G ) when H is a characteristic subgroup. In particular Inn( G )is a normal Lie subgroup of Aut( G ); the group Aut( G ) / Inn( G ) is known as thegroup of outer automorphisms of G . We note that Inn( G ) may in general notbe a closed subgroup of Aut( G ); this is the case, for example, for the semidirectproduct of R with C with respect to the action under which t ∈ R acts by( z , z ) ( e iαt z , e iβt z ) for all z , z ∈ C , with α, β fixed nonzero real numberssuch that α/β is irrational.Let Z denote the center of G and ϕ : G → Z be a (continuous) homomorphismof G into Z . Let τ : G → G be defined by τ ( x ) = xϕ ( x ) for all x ∈ G . It is easyto see that τ ∈ Aut( G ); we call it the shear automorphism , or more specifically isotropic shear automorphism (as in [20]), associated with ϕ . We note that forany continuous homomorphism ϕ as above the subgroup [ G, G ] is contained in thekernel of ϕ , and hence the associated shear automorphism τ fixes [ G, G ] pointwise.In particular, if [
G, G ] = G then there are no nontrivial shear automorphisms. Theshear automorphisms form a closed normal abelian subgroup of Aut( G ), say S .4e note that if A = { τ ∈ Aut( G ) | τ ( z ) = z for all z ∈ Z } then Aut( G ) is thesemidirect product of A and S .If H is the subgroup of G containing [ G, G ] and such that H/ [ G, G ] is themaximal compact subgroup of G/ [ G, G ], then
G/H is a vector group and the setof continuous homomorphisms ϕ of G into Z such that H ⊂ ker ϕ has the naturalstructure of a vector space, and in turn the same holds for the corresponding set ofshear automorphisms; the dimension of the vector space equals the product of thedimensions of G/H and Z . When G/ [ G, G ] is a vector group (viz. topologicallyisomorphic to R n for some n ) S is a connected algebraic subgroup of Aut( G ). Onthe other hand, when G/ [ G, G ] is compact then there are only countably manydistinct continuous homomorphisms of G into Z , and hence only countably manyshear automorphisms.For a connected semisimple Lie group G , Inn( G ) is a subgroup of finite indexin Aut( G ); in the case of a simply connected Lie group this follows from thecorresponding statement for the associated Lie algebra (cf. [53], Theorem 5.5.14),and the general case follows from the special case, since the inner automorphismsof the simply connected covering group factor to the original group G . On theother hand, for a connected nilpotent Lie group the group of outer automorphismsis always of positive dimension (see [58], Theorem 4). An example of a 3-stepsimply connected nilpotent Lie group G for which Aut( G ) = Inn( G ) · S , where S is the group of all shear automorphisms of G , is given in [26]; in particular Aut( G )is nilpotent in this case; a larger class of connected nilpotent Lie groups for whichAut( G ) is nilpotent is also described in [26].We recall here that the center of a connected Lie group G is contained in aconnected abelian Lie subgroup of G (see [53], Theorem 14.2.8, or [55], Ch. 16,Theorem 1.2). In particular the center is a compactly generated abelian group andhence has a unique maximal compact subgroup; we shall denote it by C . Being acompact abelian Lie subgroup, C is in fact the cartesian product of a torus with afinite abelian group. As the unique maximal compact subgroup of G , C is Aut( G )-invariant and hence we get a continuous homomorphism q : Aut( G ) → Aut( C ),by restriction of the automorphisms to C . Since the automorphism group of acompact abelian group is countable, it follows that Aut( G ) / ker q , where ker q isthe kernel of q , is a countable group. Aut( G ) As a closed subgroup of GL( G ), Aut( G ) is a Lie group with at most countablymany connected components (see [54] for an extensions of this to not necessarilyconnected Lie groups). It is in general not connected; for R n , n ≥
1, the automor-phism group, which is topologically isomorphic to GL( n, R ), has two connected5omponents, while for the torus T n (= R n / Z n ), n ≥
2, the automorphism groupis in fact an infinite (countable) discrete group.We shall denote by Aut ( G ) the connected component of the identity inAut( G ); it is an open (and hence also closed) subgroup of Aut( G ). We denote by c ( G ) the group of connected components, namely Aut( G ) / Aut ( G ).For a connected semisimple Lie group, since Inn( G ) is of finite index in Aut ( G )(see § ( G ) = Inn( G ). Also, for these groups the number of connectedcomponents is finite; the number is greater than one in many cases (see [57]and [71] for details on the group of connected components of Aut( G ) for simplyconnected semisimple groups G ; see also the recent paper [50] where splitting ofAut( G ) into the connected component and the component group is discussed).We note that Aut ( G ) acts trivially on the unique maximal compact sub-group C of the center of G , and hence when the image of the homomorphism q : Aut( G ) → Aut( C ), as in § c ( G ) is infinite.When C as above is the circle group, Aut( C ) is of order two, and the imageof q has at most two elements, but nevertheless c ( G ) can be infinite. A naturalinstance of this can be seen in the following: Suppose G has a closed normal sub-group H such that G/H is a torus of positive dimension; then there is a uniqueminimal subgroup with the property and it is invariant under the action of Aut( G )and hence, modifying notation, we may assume H to be Aut( G )-invariant. Thenwe have (countably) infinitely many (continuous) homomorphisms ϕ : G/H → C and for each such ϕ we have an isotropic shear automorphism of G (see § G ), and hence Aut( G )has infinitely many connected components. Thus c ( G ) is infinite in this case also.The group c ( G ) is finite if and only if Aut( G ) is “almost algebraic” as asubgroup of GL( G ); see § G ) to bealmost algebraic. Any connected Lie group is locally isomorphic to a unique (upto Lie isomorphism)simply connected Lie group, namely its universal covering group. Now let G bea simply connected Lie group and Z be the center of G . Then all connectedLie groups locally isomorphic to G are of the form G/D , where D is a discretesubgroup of Z . Moreover, for two discrete subgroups D and D of Z the Liegroups G/D and G/D are Lie isomorphic if and only if there exists a τ ∈ Aut( G )such that τ ( D ) = D . Thus the class of Lie groups (viewed up to isomorphismof Lie groups) locally isomorphic to a given connected Lie group G is in canonical6ne-one correspondence with the orbits of the action of Aut( G ) on the class ofdiscrete subgroups of it center Z , under the action induced by the Aut( G )-actionon Z , by restriction of the automorphisms to Z .For G = R n , n ≥
1, all connected Lie groups isomorphic to G are of the form R m × T n − m , with 0 ≤ m ≤ n , thus n + 1 of them altogether. For the group G ofupper triangular n × n unipotent matrices ( n ≥ G ; thus in this case there are only two non-isomorphic Lie groupslocally isomorphic to G (including the simply connected one), independently of n .On the other hand there are simply connected nilpotent Lie groups G for whichAut( G ) is a unipotent group (when viewed as a subgroup of Aut( G )) (see [26]);hence in this case the Aut( G )-action on the center has uncountably many dis-tinct orbits, and therefore there are uncountably many mutually non-isomorphicconnected Lie groups that are locally isomorphic to G .Now let G be a connected Lie group with discrete center, say Z . Then Z is finitely generated (see § Z has only countably many distinctsubgroups. Considering the indices of the subgroups in Z it can also be seenthat when Z is infinite there are infinitely many subgroups belonging to distinctorbits of the Aut( G )-action on the class of subgroups; thus in this case there arecountably infinitely many connected Lie groups locally isomorphic to G . Thisapplies in particular when G is the universal covering group of SL(2 , R ). When Z is finite the number of Lie groups locally isomorphic to G is finite, and atleast equal to the number of prime divisors of the order of Z . For simple simplyconnected Lie groups the orbits of the Aut( G )-action on the class of subgroups ofthe center have been classified completely in [48]. Let G be a connected Lie group and G be the Lie algebra of G . Let GL( G ) berealised as GL( n, R ), where n is the dimension of G , via a (vector space) basis of G . For g ∈ GL( G ) let g ij , 1 ≤ i, j ≤ n , denote the matrix entries of g and let det g denote the determinant of g . A subgroup H of GL( G ) being a real algebraic groupis equivalent to the condition that it can be expressed as the set of zeros (solutions)of a set of polynomials in g ij , 1 ≤ i, j ≤ n , and (det g ) − as the variables; in thepresent instance, the field being R , it suffices to consider a single polynomial inplace of the set of polynomials.We call a subgroup of GL( G ) almost algebraic if it is of finite index in a realalgebraic subgroup. A real algebraic subgroup is evidently a closed (Lie) subgroup7f GL( G ), and since a connected Lie group admits no proper subgroups of finiteindex it follows that any almost algebraic subgroup is an open subgroup of a realalgebraic subgroup, and in particular it is a closed subgroup of GL( G ). It is knownthat any real algebraic subgroup has only finitely many connected components (see[10], Corollary 14.5, for a precise result on the number of components), and hencethe same holds for any almost algebraic subgroup. Let G be a connected Lie group and G the Lie algebra of G . In what follows weview Aut( G ) as a subgroup of Aut( G ) and of GL( G ), via the identification intro-duced earlier ( § T n = R n / Z n for some n ≥ G (see [53], Corollary 14.1.4). Theorem 2.1. (cf. [19], [73])
Let G be a connected Lie group with Lie algebra G . Let T be a maximal torus in G . Then there exists a closed connected normalsubgroup H of G such that the following conditions hold:i) Inn( H ) is an almost algebraic subgroup of Aut( G ) , andii) Aut( G ) = Inn( H )Aut T ( G ) . The subgroup H chosen in the proof is in fact invariant under all τ ∈ Aut( G )and consequently Inn( H ) is a normal subgroup of Aut( G ). Since Inn( H ) ∩ Aut T ( G )is trivial this further implies that Aut( G ) is a semidirect product of the two sub-groups. We may also mention here that the subgroup H as chosen in [19] contains[ G, G ], the commutator subgroup of G . When Inn( G ) is an almost algebraic sub-group of Aut( G ), H as in Theorem 2.1 can be chosen to be G itself; this appliesin particular when G is an almost algebraic subgroup of GL( n, R ) for some n ≥ Aut( G ) In view of Theorem 2.1 Aut( G ) is almost algebraic if and only if Aut T ( G ) is almostalgebraic for a (and hence any) maximal torus T in G . In this respect we recallthe following specific results. Theorem 2.2. ([19], [88], [73])
Let G be a connected Lie group and G be the Liealgebra of G . Then the following statements hold.i) If the center of G does not contain a compact subgroup of positive dimension,then Aut( G ) is an almost algebraic subgroup of GL( G ) .ii) Aut ( G ) is an almost algebraic subgroup of GL( G ) ; ii) if R is the solvable radical of G and T is a maximal torus in G then Aut( G ) is almost algebraic if and only if the restrictions of all automorphisms from Aut T ( G ) to T ∩ R form a finite group of automorphisms of T ∩ R ; in particular, Aut( G ) is almost algebraic if and only if Aut( R ) is almost algebraic. Assertions (i) and (ii) above were deduced from Theorem 2.1 in [19]; (ii) wasproved earlier by D. Wigner [88], and (iii) is due to W. H. Previts and S. T. Wu[73], where improved proofs were also given for (i) and (ii).We note in particular that, in the light of Theorem 2.2(ii), Inn( G ) is containedin an almost algebraic subgroup of Aut( G ), namely Aut ( G ). This turns out tobe useful in various contexts on account of certain properties of actions of almostalgebraic subgroups (see for example §§ G be a connected Lie group. Let T be the unique maximal torus containedin the center of G . Then G/T is a connected Lie group whose center contains nonontrivial compact subgroup of positive dimension, and hence Aut(
G/T ) is analmost algebraic subgroup of Aut( G ′ ), where G ′ is the Lie algebra of G/T . Each α ∈ Aut( G ) induces an automorphism of Aut( G/T ), say ¯ α . If η : Aut( G ) → Aut( G ′ ) is the canonical quotient homomorphism, defined by η (( α ) = ¯ α for all α ∈ Aut( G ), then the image of any almost algebraic subgroup of Aut( G ) under η is almost algebraic (as the map is a restriction of a homomorphism of algebraicgroups). In particular η (Aut ( G )) is an almost algebraic subgroup. Aut( G ) almost algebraic For a class of connected Lie groups the following characterisation, incorporatinga partial converse of Theorem 2.2(i) is proved in [11]; there the issue is consideredfor all Lie groups with finitely many connected components, but we shall restricthere to when G is connected; (the general case involves some technicalities in itsformulation). Theorem 2.3. ([11])
Let G be a connected Lie group admitting a faithful finite-dimensional representation. Then Aut( G ) is almost algebraic if and only if themaximal torus contained in the center of G is of dimension at most one, and it isalso the maximal torus in the radical of G . Let G be as in Theorem 2.3 and C be the maximal torus of the center of G andsuppose that C is one-dimensional. In the context of the examples of Lie groupswith Aut( G ) having infinitely many distinct connected components discussed in § C coincides with the maximal torus in the radicalof G , then there does not exist a closed normal subgroup H such that G/H is atorus, as used in the argument there. 9t could happen that Aut( G ) may be almost algebraic (which in view of The-orem 2.2(ii) is equivalent to c ( G ) being finite), for a connected Lie group G (notadmitting a faithful finite-dimensional representation) even if its center containsa torus of dimension exceeding 1 (in fact of any given dimension). An exampleof this was indicated in [19] (page 451) and has been discussed in detail in [73].The main idea involved in the example is that Lie groups G can be constructedsuch that the central torus, though of higher dimension, is the product of one-dimensional tori, each of which is invariant under a subgroup of finite index inAut( G ). It is not clear whether there could be situations, with c ( G ) finite, forwhich this may also fail, namely with no one-dimensional compact subgroups con-tained in the center and invariant under a subgroup of finite index in Aut( G ). Itis proved in [73] (Proposition 3.1), however that for a connected Lie group G ofthe form H × T , where H is a connected Lie group and T is the one-dimensionaltorus, Aut( G ) has infinitely many connected components, and hence is not almostalgebraic. When the Lie group has additional structure, the group of automorphisms pre-serving the structure would also be of interest. In this subsection we briefly recallsome results in this respect.By a result of Hochschild and Mostow [56] if G is a connected complex affinealgebraic group then the connected component of the identity in the group A ofrational automorphisms of G is algebraic (with respect to a canonical structurearising from the associated Hoff algebra), and moreover A is itself algebraic ifeither the center of G is virtually unipotent (namely if it admits a subgroup offinite index consisting of unipotent elements) or the center of a (and hence any)maximal reductive subgroup of G is of dimension at most 1; the result may becompared with Theorem 2.2 for real Lie groups. It may be noted that the resultsof [56] are in the framework of automorphism groups of affine algebraic groupsover an algebraically closed field of characteristic zero. Some further elaborationon the theme is provided by Dong Hoon Lee [67].For the connected component of the group of complex analytic automorphismsof a faithfully representable complex analytic group, a result analogous to that ofHochschild and Mostow recalled above was proved by Chen and Wu [12]. Aut( G ) action on G Under certain conditions, a connected Lie group can be realised as a subset of R n ,or the projective space P n − , for some n ≥
2, in such a way that the automorphisms10f G are restrictions of linear or, respectively, projective transformations. We callthis linearisation of the Aut( G )-action on G . We next discuss various results inthis respect.Let G be a connected Lie group. By an affine automorphism of G we mean atransformation of the form T g ◦ τ where τ ∈ Aut( G ), and T g is a left translationby an element g in G , namely T g ( x ) = gx for all x ∈ G . We denote by Aff( G ) thegroup of all affine automorphisms. We identify G canonically as a subgroup ofAff( G ), identifying g ∈ G with T g as above. Then Aff( G ) is a semidirect product ofAut( G ) with G , with G as the normal subgroup, and we shall consider it equippedwith the Cartesian product topology. We shall denote by Aff ( G ) the connectedcomponent of the identity in Aff( G ).Let A and B be the Lie algebras of Aut( G ) and Aff( G ) respectively. LetAd : Aff( G ) → GL( B ) be the adjoint representation of Aff( G ). Let a be thedimensions of Aut( G ) and V = ∧ a B , the vector space of a -th exteriors over B .Let ρ : Aff( G ) → GL( V ) be the representation arising as the a -th exterior power ofAd; we call ρ the linearising representation for G (for reasons that would becomeclear below). Let L be the vector subspace ∧ a A of V = ∧ a B ; since a is thedimension of A , L is a one-dimensional subspace.Consider the (linear) action of Aff( G ) on V via the representation ρ . Let P = P ( V ) denote the corresponding projective space, consisting of all lines (one-dimensional subspaces) in V , equipped with its usual topology. The Aff( G )-actionon V induces an action of Aff( G ) on P . We have the corresponding actionsof Aut( G ) and G by restriction of the action ( G being viewed as the group oftranslations as above). G in a projective space Let p denote the point of P corresponding to the line L as above. Let S be thestabiliser of p under the action of G . Consider any g ∈ S . Then L is invariantunder the action of g on V and this means that the subspace A is invariant underthe action of g on B . In turn we get that Aut ( G ) is normalised by g in Aff( G ),and since Aff ( G ) is a semidirect product of Aut ( G ) and G this implies that g iscontained in the center of Aff ( G ). Conversely it is easy to see that every elementof the center of Aff ( G ) fixes p . Thus S coincides with the center of Aff ( G ). Wenote also that S = { g ∈ G | τ ( g ) = g for all τ ∈ Aut ( G ) } . The orbit map g gp of G into P induces a canonical continuous bijection j of G/S onto its image in P , defined by j ( gS ) = gp for all g ∈ G . When S is trivial j defines a continuous embedding of G into P . We note that for any11 ∈ Aut( G ) and g ∈ G we have τ ( g ) p = τ gτ − p = τ gp = τ ( gp ). Thus themap j is equivariant with respect to the actions of Aut( G ) on G/S and P , viz. j ( τ ( gS )) = τ j ( gS ) for all g ∈ G and τ ∈ Aut( G ). Thus when S is trivial theorbits of the Aut( G )-action on G/S are in one-one correspondence in a naturalway with orbits of the Aut( G )-action on P that are contained in the image of G/S .Recall that Aut ( G ) is an almost algebraic subgroup of GL( G ) (see Theo-rem 2.2(ii)). From this it can be deduced that the restriction of the representa-tion ρ as above to Aut ( G ) is an algebraic representation, viz. restriction of analgebraic homomorphism of algebraic groups. Hence every orbit of Aut ( G ) on P is locally closed, namely open in its closure (see [72], Lemma 1.22, for instance).Thus the above argument shows the following. Theorem 2.4.
Let G be a connected Lie group and S be the center of Aff ( G ) .Then every orbit of the action of Aut ( G ) on G/S is locally closed (viz. open inits closure).
The subgroup S , which is noted to be contained in the center of G , containsthe maximal compact subgroup of the center. In particular if S is trivial then byTheorem 2.2(i) Aut ( G ) has finite index in Aut( G ) and hence the above theoremimplies that all orbits of Aut( G ) on G are locally closed.We recall here that the condition of the orbits being locally closed is wellstudied in a wider context and is equivalent to a variety of other conditions ofinterest; see [44]; (see also [42] for a more general result). It represents in variousways the opposite extreme of the action being ergodic. In analogy with the embedding of G as a subset of P (modulo the subgroup S as above) we can also get an embedding of G in the vector space V as definedabove, as follows. Let the notation be as above and let v be a nonzero point of L . We note that S as above is also the stabilizer of v under the G -action on V ,since if g fixes v it fixes p , and if it fixes p then we have τ ( g ) = τ gτ − = g for all τ ∈ Aut ( G ) and hence it fixes v . As L is invariant under the actionof Aut( G ) we get a continuous homomorphism s : Aut( G ) → R ∗ such that forall τ ∈ Aut( G ), τ ( v ) = s ( τ ) v . For any g ∈ G and τ ∈ Aut( G ) we have τ ( g ) v = ( τ gτ − ) v = τ g ( τ − ( v )) = s ( τ − ) τ gv . Consider the (linear) actionof Aut ( G ) on V such that τ ∈ Aut ( G ) acts by v ∈ V s ( τ − ) τ v . Thenthe Aut ( G )-orbits on G/S are in canonical correspondence with the orbits on V under this Aut ( G )-action. Under this action also the orbits are locally closed, byalgebraic group considerations. 12 .8 Algebraicity of stabilizers Let G be a connected Lie group and ρ : Aff( G ) → GL( V ) be the correspondinglinearising representation as in § ( G ) is an almost algebraicsubgroup of GL( G ), G being the Lie algebra of G , and that the restriction of ρ toAut ( G ) is the restriction of a homomorphism of algebraic groups. In particular,for any v ∈ V the stabiliser { τ ∈ Aut ( G ) | τ ( v ) = v } is an almost algebraicsubgroup of Aut ( G ) (algebraic subgroup if Aut ( G ) is algebraic). Let v be asin § S , as before, be the center of Aff ( G ), which is the stabiliser of v under the action as in § g ∈ G thesubgroup { τ ∈ Aut ( G ) | τ ( gS ) = gS } is almost algebraic. In particular for aconnected Lie group G such that the center of Aff ( G ) is trivial, for all g ∈ G thestabiliser { τ ∈ Aut ( G ) | τ ( g ) = g } of g under the action of Aut ( G ) is an almostalgebraic subgroup; this holds in particular when the center of G is trivial.Let G be a connected Lie group and for g ∈ G let S ( g ) denote the stabiliser { τ ∈ Aut( G ) | τ ( g ) = g } . It is shown in [41] that when G is a simply connectedsolvable Lie group, S ( g ) is an algebraic subgroup for all g ∈ G . In the generalcase it is shown that the connected component of the identity in S ( g ) is an almostalgebraic subgroup for all g which are of the form exp ξ for some ξ in the Liealgebra of G , exp being the exponential map associated with G . It is noted that S ( g ) itself need not be an algebraic subgroup, as may be seen in the case when G is the universal covering group of SL(2 , R ) and g is one of the generators of thecenter of G (the latter is an infinite cyclic subgroup); in this case S ( g ) = Inn( G ),and it is of index 2 in Aut( G ), but not an algebraic subgroup.For elements z contained in the center of G it is proved in [47] (see also [19])that the connected component of the identity in S ( z ) is an almost algebraic sub-group, and if G has no compact central subgroup of positive dimension then S ( z )itself is almost algebraic. It is however not true that S ( z ) is almost algebraicwhenever Aut( G ) is almost algebraic; a counterexample in this respect may befound in [73] (p. 432).We note in particular that if A is the subgroup of Aut( G ) consisting of allautomorphisms which fix the center pointwise then the connected component ofthe identity in A is an almost algebraic subgroup, and if G has no compact centralsubgroup of positive dimension then A is almost algebraic. It is well-known that an automorphism α of the n -dimensional torus T n , n ≥ § dα has no eigenvalue whichis a root of unity. These automorphisms constitute some of the basic examplesin ergodic theory and topological dynamics, and have been much studied fordetailed properties from the point of view of the topics mentioned (see [14], [87],for instance; see also [79] and [22] for generalisations). In this section we discussactions of subgroups of Aut( G ) with a dense orbit on G . Aut( G ) -actions with dense orbits Let G be a connected abelian Lie group. Then it has the form R m × T n for some m, n ≥ G )-action on G has denseorbits, except when ( m, n ) = (0 ,
1) (viz. when G is the circle group); if n = 0 thecomplement of 0 in R m is a single orbit. If m ≥ ( G ) alsohas a dense orbit on G , as may be seen using the isotropic shear automorphisms(see § R m into T n .Next let G be a 2-step connected nilpotent Lie group, namely [ G, G ] is con-tained in the center of G . Let Z denote the center of G . Then G/Z is simplyconnected, and hence is Lie isomorphic to R n for some n . The Aut( G )-actionon G factors to an action on G/Z . Using shear automorphisms (see § G )-action on G has a dense orbit on G if and only if theAut( G )-action on G/Z has a dense orbit. The latter condition holds in particularif G is a free 2-step simply connected Lie group; it may be recalled that G is calleda free 2-step nilpotent Lie group if its Lie algebra is of the form V ⊕ ∧ V , with V a finite-dimensional vector space over R , and the Lie product is generated bythe relations [ u, v ] = u ∧ v for all u, v ∈ V , and [ u, v ∧ w ] = 0 for all u, v, w ∈ V .When G is a free 2-step simply connected nilpotent Lie group the quotient G/Z as above corresponds to the vector space V and every nonsingular automorphismof V is the factor of a τ ∈ Aut( G ), which leads to the observation as above.For a general simply connected 2-step nilpotent Lie group G the Lie algebra G of G can be expressed as ( V ⊕ ∧ V ) /W , where V is the vector space G/Z ( Z being the center of G ) and W is a vector subspace of ∧ V ; V ⊕ ∧ V is the free2-step nilpotent Lie algebra with the structure as above, and any vector subspaceof ∧ V is a Lie ideal in the Lie algebra (see [5], where the example is discussedin a different context). It is easy to see that in this case the image of Aut( G ) inGL( V ), under the map associating to each automorphism its factor on V = G/Z ,is the subgroup, say I ( W ), consisting of g ∈ GL( V ) such that the correspondingexterior transformation ∧ ( g ) of ∧ V leaves the subspace W invariant. It can beseen that I ( W ) has an open dense orbit on V for various choices of W , and inthese cases by the argument as above Aut( G ) has open dense orbit on G . Forexample, if e , . . . , e n , n ≥ V , then this is readily seen to hold14f W is the subspace spanned by e ∧ e or, more generally, by sets of the form { e ∧ e , e ∧ e , . . . , e k − ∧ e k } , for n ≥ k .Along the lines of the above arguments it can be seen that for any k ≥ G is a free k -step simply connected Lie group then the action of Aut( G ) on G hasan open dense orbit. Also, given a simply connected k -step nilpotent Lie group H there exists simply connected k + 1-step nilpotent Lie group G such that H isLie isomorphic to G/Z , where Z is the center of G , and every automorphism of H is a factor of an automorphism of G . Therefore using the examples as above onecan also construct examples, for any k ≥
2, of simply connected k -step nilpotentLie groups such that the Aut( G )-action on G has an open dense orbit.In the converse direction we have the following. Theorem 3.1.
Let G be a connected Lie group. Suppose that there exists g ∈ G such that the closure of the Aut( G ) -orbit of g has positive Haar measure in G .Then G is a nilpotent Lie group. A weaker form of this, in which it was assumed that the orbit is dense in G ,was proved in [23] (Theorem 2.1), but the same argument is readily seen to yieldthe stronger assertion as above. By a process of approximation, Theorem 2.1 in[23] was extended to all finite-dimensional connected locally compact groups, andin the same way one can also get that the assertion in Theorem 3.1 holds also forall finite-dimensional connected locally compact groups.A nilpotent Lie group G need not always have dense orbits under the action ofAut( G ). Examples of 2-step simply connected nilpotent Lie groups with no denseorbits under the Aut( G )-action are exhibited in [26]. There are also examples in[26] of 3-step simply connected nilpotent Lie groups for which Aut( G ) is a unipo-tent group, namely when Aut( G ) is viewed a subgroup of GL( G ) all its elementsare unipotent linear transformation, and consequently all orbits of Aut( G ) on G are closed, and hence lower dimensional, submanifolds of G .There has been a detailed study of groups in which Aut( G ) has only finitelymany, or countably many, orbits, in the broader context of locally compact groups,and also abstract groups. The results in this respect for connected Lie groups willbe discussed in § Aut( G ) with dense orbits Theorem 3.1 implies in particular that there is no connected Lie group G forwhich Aut( G ) has a one-parameter subgroup whose action on G has a denseorbit; by the theorem, a Lie group G with that property would be nilpotent andsince it has to be noncompact, going to a quotient we get that for some n ≥ n, R ) has a one-parameter subgroup acting on R n with a dense orbit, butsimple considerations from linear algebra rule this out.On R n viewed as C n , n ≥
2, we have linear actions of C n − × C ∗ , whichhave an open dense orbit: the action of ( z , . . . , z n − , z ), where z , . . . , z n − ∈ C and z ∈ C ∗ , is defined on the standard basis vectors { e , . . . , e n } by e j ze j for j = 1 , . . . , n − e n P n − j =1 z j e j + ze n . The same also holds for (outer)cartesian products of such actions, and in particular we have linear actions of C ∗ n on R n , with dense orbits. Since for n ≥ R n +1 as a dense subgroupof C ∗ n , we get linear actions R n +1 on R n admitting dense (but not open) orbits;we note that in analogy with the above we can get linear actions of R ∗ n on R n with open dense orbits, but they do not yield actions of R n with dense orbits.It can be seen using the Jordan canonical form that there is no linear action of R admitting dense orbits, and hence there is no action of R on any Lie groupadmitting dense orbits, by an argument as above, using Theorem 3.1.On R n , apart from GL( n, R ), various proper subgroups act transitively onthe complement of { } ; e.g. S · O ( n, R ) where S is the subgroup consisting ofnonzero scalar matrices and O ( n, R ) is the orthogonal group, or the symplecticgroup Sp( n, R ) for even n . Also if O ( p, q ) is the orthogonal group of a quadraticform of signature ( p, q ) on R n , n = p + q then the action of S · O ( p, q ), with S asabove, on R n has an open dense orbit (the complement consists of the set of zerosof the quadratic form, which is a proper algebraic subvariety of R n ).Subgroups of GL( n, R ) acting with an open dense orbit have been a subjectof much interest in another context; R n together with such a subgroup is called apre-homogeneous vector space; the reader is referred to [63] for details. Analysis of the issue of dense orbits was inspired by a question raised by P. R.Halmos in his classic book on Ergodic Theory ([51], page 29) as to whether a non-compact locally compact group can admit a (continuous) automorphism which isergodic with respect to the Haar measure of the group, namely such that thereis no measurable set invariant under the automorphism such that both the setand its complement have positive Haar measure (see § C Z where C is a compact nonabelian group, for which theshift automorphism has dense orbits, a compact connected Lie group admits suchan automorphism only if it is a torus of dimension n ≥
2. For a compact semisim-ple Lie group G , Inn( G ) is a subgroup of finite index in Aut( G ) and the orbitsof Aut( G ) on G are closed submanifolds of dimension less than that of G , and inparticular not dense in G ; a general compact connected Lie group has a simpleLie group as a factor and hence the preceding conclusion holds in this generalityalso. Z d -actions The analogue of Halmos’ question for Z d -actions, namely the multi-parameter casewith d ≥
1, and more generally actions of abelian groups of automorphisms, wasconsidered in [24], where the following is proved.
Theorem 3.2.
Let G be a connected Lie group. Suppose that there exists anabelian subgroup H of Aut( G ) such that the H -action on G has a dense orbit.Then there exists a compact subgroup C contained in the center of G such that G/C is topologically isomorphic to R n for some n ≥ ; in particular G is atwo-step nilpotent Lie group. If moreover the H -action leaves invariant the Haarmeasure on G then G is a torus. Recall that the n -dimensional torus T n , where n ≥
2, admits automorphismswith a dense orbit, and one can find such an automorphism contained in a sub-group A of Aut( T n ) ≈ GL( n, Z ) which is isomorphic to Z d for d ≤ n −
1. Thisgives examples of subgroups A of Aut( T n ) isomorphic to Z d for d ≤ n −
1, actingwith a dense orbit. Conversely every subgroup A of Aut( T n ) isomorphic to Z d and acting with a dense orbit contains an (individual) automorphism which hasa dense orbit; see [7]; see also [76] for analogous results in a more general settingof automorphisms of general compact abelian groups.It is easy to see that we have a Z -action on R with a dense orbit; the automor-phisms defined respectively by multiplication by e α and − e β , α, β >
0, generatesuch an action when α/β is irrational. More generally, G = R n × T m admits a Z d -action with a dense orbit if any only if m = 1 and d ≥ ( n + 2) /
2; see [24]for details. Examples of nonabelian two-step nilpotent Lie groups G admitting Z -actions with a dense orbit are given in [24].17t may be mentioned here that Theorem 3.2 is extended in [38] to actions ongeneral locally compact groups G , where it is concluded that under the analogouscondition there exists a compact normal subgroup C such that the quotient G/C is a (finite) product of locally compact fields of characteristic zero.In line with the above it would be interesting to know about Lie groups ad-mitting actions by nilpotent or, more generally, solvable groups of automorphismswith a dense orbit.
Consider a connected Lie group G with a Lie subgroup H of Aut( G ) such thatthe H -action on G has an open dense orbit; e.g. R n , n ≥
1, and H = SL( n, R )– see § G is nilpotent). Let g ∈ G be such that the H -orbit is open and dense in G . Let L be the stabiliser of g under the H -action, viz. L = { τ ∈ Aut( G ) | τ ( g ) = g } . Then it is easy to seethat for a subgroup Γ of H the Γ-action on G has a dense in G if and only if the L -action on H/ Γ has a dense orbit; this phenomenon is known as “duality” - seefor instance [6], [22] for some details). In certain situations, such as when Γ is alattice in H (viz. H/ Γ admits a finite measure invariant under the action of H on the left), the question of whether the action of a subgroup has dense orbits on H/ Γ is amenable via techniques of ergodic theory.For G = R n , n ≥
2, we have Aut( G ) ≈ GL( n, R ), and there exist many discretesubgroups of the latter whose action on R n admits dense orbits. Let H = SL ( n, R )and Γ be a lattice in SL( n, R ). We choose g as e where { e , . . . , e n } is thecoordinate basis of R n and let L be its stabiliser. It is known that this subgroupacts ergodically on H/ Γ and hence as noted above the Γ-action on R n has a denseorbit. This applies in particular to the subgroup SL( n, Z ), consisting of integralunimodular matrices, which is indeed a lattice in SL( n, R ) (see [70], Ch. 7 or [75],Ch. 10). We shall discuss more about the orbits of these in the next section.There are also natural examples of discrete subgroups of SL( n, R ) other thanlattices which have dense orbits under the action on R n . For example if n is evenand Γ is a lattice in the symplectic group Sp( n, R ) then it has dense orbits on R n ; stronger statements analogous to those for lattices in SL( n, R ) are possiblebut we shall not go into the details. There are also other examples, arising fromhyperbolic geometry. Let Γ be the fundamental group of a surface of constantnegative curvature whose associated geodesic flow is ergodic. Then Γ may beviewed canonically as a subgroup of PSL(2 , R ) = SL(2 , R ) / {± I } , where I denotesthe identity matrix, and if ˜Γ is the lift of Γ in SL(2 , R ), then the action of ˜Γ on R has dense orbits; the action is ergodic with respect to the Lebesgue measure (see[6] for some details; see § R n one may consider other nilpotent connected Lie groups G whichadmit an open dense orbit under the action of Aut( G ). It would be interesting toknow analogous results for discrete groups of automorphisms of other connectednilpotent Lie groups, which however does not seem to be considered in the liter-ature. In most of the cases considered in the earlier subsections where we conclude ex-istence of a dense orbit, not all orbits may be dense, and in general there is nogood description possible of the ones that are not dense. However in certain casesa more complete description is possible.Expressing T n as R n / Z n , Aut( T n ) can be realised as GL( n, Z ), the group ofall n × n matrices with integer entries and determinant ±
1. For v ∈ R n whosecoordinates with respect to the standard basis (generating Z n ) are rational theorbits under the Aut( T n )-action are easily seen to be finite. It turns out, and nottoo hard to prove (see [30], for instance) that conversely for any v at least one ofwhose coordinates is irrational the Aut( T n )-orbit is dense.For the case of R n , n ≥ n, R )then there exist dense orbits, by duality and ergodicity considerations. In factin this case it is possible to describe the dense orbits precisely. If SL( n, R ) / Γis compact (viz. if Γ is a “uniform” lattice) then the orbit of every non-zeropoint in R n is dense in R n . When SL( n, R ) / Γ is noncompact (but a lattice) theset of points whose orbits are not dense is contained in a union of countablymany lines (one-dimensional vector subspaces) in R n . In the case of SL( n, Z ) theexceptional lines involved are precisely those passing through points in R n withrational coordinates (see [35] for more general results in this direction). Theseresults are consequences of the study of flows on homogeneous spaces which hasbeen a much studied topic in the recent decades, thanks to the work of MarinaRatner on invariant measures of unipotent flows. We shall not go into details onthe topic here; the interested reader is referred to the expository works [22] and[64] and other references there, for exploration of the topic. The result in thespecial case of SL( n, Z ) recalled above was first proved in [15] (see also [34] for astrengthening in another direction).As in the case of the issues in the previous section, it would be interestingto know results analogous to the above for discrete groups of automorphisms ofother nilpotent connected Lie groups. 19 .7 Aut( G ) -actions with few orbits Observe that when G = R n , n ∈ N , Aut( G ) is GL( n, R ) and the action is transitiveon the complement of the zero element. Thus the action of Aut( G ) on G has onlytwo orbits. The question as to when there can be only finitely many, or countablymany, orbits has attracted attention, not only for Lie groups, but in the generalcontext of locally compact groups; we shall indicate some of the results in thatgenerality, giving references, but our focus shall be on Lie groups. Theorem 3.3.
Let G be a locally compact group. Suppose that the action of Aut( G ) on G has only countably many orbits. Then the connected component G of the identity in G is a simply connected nilpotent Lie group. Moreover, thenumber of Aut( G ) -orbits in G is finite and one of them is an open orbit. The first statement in the theorem was proved in [84]. We note that for Liegroups it can be deduced from Theorem 3.1 (which is a generalised version ofa theorem from [23]); since under the condition in the hypothesis at least oneof the orbits has to be of positive measure Theorem 3.1 yields that the groupis nilpotent, but on the other hand the condition also implies that there is nocompact subgroup of positive dimension contained in the center, so G must infact be a simply connected nilpotent Lie group. It is proved in [85] (Theorem 6.3)that under the condition in the hypothesis one of the orbits is open. Moreover, as G is a simply connected nilpotent Lie group, the Aut( G )-orbits are in canonicalone-one correspondence with Aut( G )-orbits on G via the exponential map, where G is the Lie algebra of G , and the latter being an action of a real algebraic group,the cardinality of the orbits can be countable only if it is finite and one of theorbits is open.Now let G be a connected Lie group such that the Aut( G )-action on G hasonly finitely many orbits. Since the identity element is fixed, there are at leasttwo orbits; the group is said to be homogeneous if there are only two orbits. Thegroups R n , n ≥
1, are homogeneous, and they are also readily seen to be theonly connected abelian Lie groups for which the Aut( G )-action has only finitelymany orbits. It turns out that R n , n ≥
1, are in fact the only connected locallycompact groups that are homogeneous (see [82], Theorem 6.4); all (not necessarilyconnected) homogeneous locally compact groups have also been determined in [82]; K n , where K is the field of p -adic numbers, with p is a prime, or Q (with thediscrete topology), and n ∈ N , are some of the other examples of homogeneousgroups.A locally compact group is said to be almost homogeneous if the Aut( G )-actionon G has 3 orbits. The class of groups with the property has been studied in [84]and [85]. An almost homogeneous Lie group is a Heisenberg group, namely a20roup defined on V ⊕ Z , where V and Z are vector spaces, with the productdefined by ( v, x ) · ( w, y ) = ( v + w, x + y + h v, w i ), where h· , ·i is an alternatingnon-degenerate bilinear form over V with values in Z ; (we note that here thegroup structure is viewed via identification with the corresponding Lie algebra,and the factor is introduced so that h v, w i is the Lie bracket of v and w ).Moreover, the pair of dimensions of the vector spaces V and Z are either of theform (2 n, , (4 n, , (4 n, n ∈ N , or one of (3 , , (6 , , (7 , , (8 , , (8 , , G is a k -step simply connected nilpotent Lie group thenthe number of Aut( G )-orbits on G is at least k + 1. If G is one of the Heisenberggroups as above then for any r ≥
1, the action of Aut( G r ) on G r has 2 r + 1 orbits(cf. [85], Proposition 6.8). Thus there exist nilpotent Lie groups with arbitrarilylarge finite number of orbits under the action of the respective automorphismgroup. We refer the reader to [85], and other references there, for further detailsand also open problems on this theme, for general (not necessarily connected)locally compact groups. In this section we discuss various aspects of ergodic theory of actions on Lie groupsby groups of automorphisms.
We begin by briefly recalling some definitions and conventions which will be fol-lowed throughout. By a measure we shall always mean a σ -finite measure. Givena measure space ( X, M ), an automorphism τ : X → X is said measurable if τ − ( E ) ∈ M for all E ∈ M ; a measure µ on ( X, M ) is said to be invariant undera measurable automorphism τ if µ ( τ − ( E )) = µ ( E ) for all E ∈ M , and it is saidto be quasi-invariant if, for E ∈ M , µ ( τ − ( E )) = 0 if and only if µ ( E ) = 0; ameasure is said to be invariant or, respectively, quasi-invariant, under a groupof measurable automorphisms if it has the property with respect to the actionof each of the automorphisms from the group. A measure which is invariant orquasi-invariant with respect to an action is said to be ergodic with respect to theaction if there are no two disjoint measurable subsets invariant under the action,each with positive measure. 21wo measures on a measurable space X are said to be equivalent if they havethe same sets of measure 0. It is easy to see that given a measure which is quasi-invariant under an action there exists a finite measure equivalent to it, whichis also quasi-invariant; in particular, given an infinite invariant measure thereexists a finite quasi-invariant measure equivalent to it (which may however not beinvariant under the action).A measure µ on G is called a probability measure if µ ( G ) = 1. We denote by P ( G ) the space of probability measures on G . For actions of a locally compactsecond countable group every quasi-invariant probability measure can be “decom-posed as a continuous sum” of ergodic quasi-invariant probability measures in acanonical way, and hence it suffices in many respects to understand the ergodicquasi-invariant measures (see, for instance, [70], Theorem 14.4.3).In our context the measure space structure will always be with respect to theBorel σ -algebra of the topological space in question. When G is the torus T n , n ≥
2, the Haar measure is invariant under the action ofAut( G ), which is an infinite discrete group. More generally, if G is a connected Liegroup such that the center contains a torus of positive dimension, then the Haarmeasure of the maximal torus of the center, viewed canonically as a measure on G ,is invariant under Aut( G ), which can be a group with infinitely many connectedcomponents (see § G ) the situation is quite in contrast, as will be seenin Theorem 4.1 below.Let G be a connected Lie group. For µ ∈ P ( G ) we denote by supp µ thesupport of µ , namely the smallest closed subset of G whose complement is of µ -measure 0. Let I ( µ ) = { τ | µ is invariant under the action of τ } , and J ( µ ) = { τ ∈ Aut( G ) | τ ( g ) = g for all g ∈ supp µ } . Then it can be seen that I ( µ ) and J ( µ ) are both closed subgroups of Aut( G ) and J ( µ ) is a normal subgroup of I ( µ ). It turns out that when G has no compactsubgroup of positive dimension contained in the center, the quotient I ( µ ) /J ( µ ) iscompact. In fact we have the following. Theorem 4.1. (cf. [20])
Let G be a connected Lie group. Let A be an almostalgebraic subgroup of Aut( G ) . Then for any µ ∈ P ( G ) , ( I ( µ ) ∩ A ) / ( J ( µ ) ∩ A ) iscompact. G such that Aut( G ) is almost algebraic (seeTheorem 3.1) then I ( µ ) /J ( µ ) is compact and in turn for any x ∈ supp µ the I ( µ )-orbit of x in G is compact; thus supp µ can be expressed as a disjoint union ofcompact orbits of I ( µ ).We recall that if G is a compact Lie group then the Haar measure of G isa finite measure invariant under all automorphisms. On the other hand when G = R n for some n ≥
1, the only finite invariant measure is the point mass at 0(see Corollary 4.2 below). The general situation is, in a sense, a mix of the twokinds of situations.From Theorem 4.1 one can deduce the following, by going modulo the maxi-mal compact subgroup contained in the center and applying Theorem 3.1 on thequotient.
Corollary 4.2.
Let G be a connected Lie group and H be a subgroup of Aut( G ) .Let µ be a finite H -invariant measure on G . Then for any g ∈ supp µ the H -orbitof g is contained in a compact subset of G . In particular if H does not have anorbit other than that of the identity which is contained in a compact subset, thenthe point mass at the identity is the only H -invariant probability measure on G . For G = R n , n ≥
1, Therorem 4.1 implies in particular the following result,which was proved earlier in [18].
Theorem 4.3.
For µ ∈ P ( R n ) , I ( µ ) is an algebraic subgroup of Aut( R n ) =GL( n, R ) . Analogous results are also proved in [18] for projective transformations. Theapproach involves the study of non-wandering points of the transformations, whichhas been discussed in further detail in [28].
We recall here the following result concerning convolution powers of probabilitymeasures invariant under the action of a compact subgroup of positive dimension.For a probability measure µ and k ∈ N we denote by µ k the k -fold convolutionpower of µ . Theorem 4.4.
Let µ be a probability measure on R n , n ≥ . Suppose that µ is invariant under the action of a compact connected subgroup K of Aut( G ) =GL( n, R ) of positive dimension. Then one of the following holds:i) µ n is absolutely continuous with respect to the Lebesgue measure on R n ;ii) there exists an affine subspace W of R n such that µ ( W ) > . K -action has no nonzero fixed point and there is no proper vector subspace U with µ ( U ) > U can also be stipulated to be invariant), so under theseassumptions µ n is absolutely continuous with respect to the Lebesgue measure;this is the formulation of Theorem 3.2 in [29].It would be interesting to have a suitable analogue of the above theorem foractions on a general connected Lie group G by compact subgroups of positivedimension in Aut( G ), (with the Lebesgue measure replaced by the Haar measureof G ). The Lebesgue measure on R n is invariant under the action of a large subgroupof Aut( G ) = GL( n, R ), namely SL( n, R ), the special linear group, but not thewhole of Aut( G ). On the other hand if G is a connected semisimple Lie groupthen Inn( G ) is of finite index in Aut( G ), and as G is unimodular it follows thatthe Haar measure is invariant under the action of Aut( G ). In many Lie groups ofcommon occurrence it is readily possible to determine whether the Haar measure isinvariant under all automorphisms, but there does not seem to be any convenientcriterion in the literature to test it.Let G be a connected Lie group and let S denote the subgroup which is thecenter of Aff ( G ), when G is viewed canonically as a subgroup of Aff( G ) (see § ( G ), and more generally of anyalmost algebraic subgroup A of Aut( G ), on G/S are locally closed. By a theoremof Effros [42] this implies that for every measure on G which is quasi-invariant andergodic under the action of an almost algebraic subgroup A of Aut( G ), there existsan A -orbit O such that the complement of O has measure 0 (that is, the measureis “supported” on O , except that the latter is a locally closed subset which maynot be closed); in particular this applies to any infinite measure invariant underthe action of an almost algebraic subgroup A of Aut( G ) as above. We recall that for an action of a group on a locally compact second countablespace, given an ergodic quasi-invariant measure µ , for almost all points in supp µ ,the orbit is dense in supp µ . Together with the results on existence of dense orbitsin § Theorem 4.5.
Let G be a connected Lie group and let λ be a Haar measure of . Then λ is quasi-invariant under the action of Aut( G ) . Moreover the followingholds:i) if λ is ergodic with respect to the Aut( G ) -action on G then G is a nilpotentLie group.ii) if λ is ergodic under the action of an abelian subgroup of Aut( G ) then G isa two-step nilpotent Lie group, with [ G, G ] compact.iii) if λ is ergodic under the action of some α ∈ Aut( G ) then G is a torus.iv) if λ is ergodic under the action of an almost algebraic subgroup A of Aut( G ) then the A -action on G/C , where C is the maximal compact subgroup of G , hasan open dense orbit. The first three statements are straightforward consequences of results from § { ¯ α | α ∈ A } is an almost algebraic subgroup of Aut( G ′ ),where G ′ is the Lie algebra of G/C , and hence its orbits on G ′ are locally closed;as G/C is a simply connected nilpotent Lie group the exponential map is a home-omorphism and hence the preceding conclusion implies that the A -orbits on G/C are locally closed, and in particular an orbit which is dense is also open in
G/C . Let G be a connected Lie group and consider a (Borel-measurable) action of G on astandard Borel space X . By the stabiliser of a point x ∈ X we mean the subgroup { g ∈ G | gx = x } , and we denote it by G x . Each G x is a closed subgroup of G (see[86], Corollary 8.8). When the action is transitive, namely when the whole of X is a single orbit, then the stabilisers of any two points in X are conjugate to eachother. One may wonder to what extent this generalises to a general ergodic action,with respect to a measure which is invariant or quasi-invariant under the action of G . In this respect the following is known from [25]; the results in [25] strengthenthose from [45] in the case of Lie groups, while in [45] the issue of conjugacy ofthe stabilisers is considered in the wider framework of all locally compact groups. Theorem 4.6.
Let G be a connected Lie group acting on a standard Borel space X . Let µ be a measure on X which is quasi-invariant and ergodic with respectto the action of G . Suppose also that for all x ∈ X the stabiliser G x of x hasonly finitely many connected components. Then there exists a subset N of X with µ ( N ) = 0 such that for x, y ∈ X \ N there exists α ∈ Aut ( G ) such that α ( G x ) = G y ; in particular G x and G y are topologically isomorphic to each other. This is a variation of Corollary 5.2 in [25], where the ergodicity conditioninvolved is formulated in terms of a σ -ideal of null sets, rather than a quasi-25nvariant measure. Without the assumption of G x having only finitely manyconnected components it is proved that for x, y in the complement of a null set N , the connected components of the identity in G x and G y are topologicallyisomorphic to each other ([25], Corollary 5.7). The proofs of these results arebased on consideration of the action of Aut( G ) on the space of all closed subgroupsof G , the latter being equipped with the Fell topology, and the orbits or stabilisers G x , x ∈ X , under the action. The argument depends on the fact Aut ( G ) is analmost algebraic subgroup of GL( G ) containing Inn( G ), G being the Lie algebraof G (see Theorem 2.2(ii)); the proof shows that automorphisms α as in theconclusion of the theorem may in fact be chosen to be from the smallest almostalgebraic subgroup containing Inn( G ), and in particular if Inn( G ) is an almostalgebraic subgroup of GL( G ) then stabilisers of almost all points are conjugatesto each other. Other conditions under which the stabilisers of almost all pointsmay be concluded to be actually conjugate in G , and also examples for which itdoes not hold, are discussed in [25]. We shall not go into the details of these here. Aut( G ) -actions In this section we discuss certain results on dynamics in which some features ofthe actions of Aut( G ) on G described in earlier sections play a role. Let G be a connected Lie group and consider a continuous action of G on acompact Hausdorff space X . Here we recall a topological analogue of Theorem 4.6.As before we denote by G x the stabiliser of x under the action in question and wedenote by G x the connected component of the identity in G x . We recall that anaction is said to be minimal if there is no proper nonempty closed subset invariantunder the action. The following is proved in [27], Proposition 3.1. Theorem 5.1.
Let G be a connected Lie group acting continuously on a locallycompact Hausdorff space X . Suppose that there exists x ∈ X such that the orbit of x is dense in X . Then there exists an open dense subset Y of X such that for all y ∈ Y there exists α ∈ Aut ( G ) such that α ( G x ) = G y . If the action is minimalthen G x , x ∈ X , are Lie isomorphic to each other. Further conditions which ensure G x , x ∈ X , being conjugate in G are discussedin [27]. The results of [27] generalise earlier results of C. C. Moore and G. Stuckproved for special classes of Lie groups (see [27] for details).26 .2 Anosov automorphisms Anosov diffeomorphisms have played an important role in the study of differen-tiable dynamical systems; we shall not go into much detail here on the issue –the reader is referred to Smale’s expository article [81] for a perspective on thetopic. Hyperbolic automorphisms of tori T n , n ≥
2, namely automorphisms α such that dα has no eigenvalue (including complex) of absolute value 1, serveas the simplest examples of Anosov automorphisms. In [81] Smale described anexample, due to A. Borel, of a non-toral compact nilmanifold admitting Anosovautomorphisms; a nilmanifold is a homogeneous space of the form N/ Γ where N is a simply connected nilpotent Lie group and Γ is discrete subgroup, and anAnosov automorphism of N/ Γ is the quotient on N/ Γ of an automorphism α of N such that α (Γ) = Γ and dα has no eigenvalue of absolute value 1. Such a sys-tem can have a nontrivial finite group of symmetries and factoring through themleads to some further examples on what are called infra-nilmanifolds, known asAnosov automorphisms of infra-nilmanifolds. Smale conjectured in [81] that allAnosov diffeomorphisms are topologically equivalent to Anosov automorphismson infra-nilmanifolds.A broader class of Anosov automorphisms of nilmanifolds, which includes alsothe example of Borel referred to above, was introduced by L. Auslander andJ. Scheuneman [5]. Their approach involves analysing Aut( N ) for certain simplyconnected nilpotent Lie groups N to produce examples of Anosov automorphismsof compact homogeneous spaces of N/ Γ for certain discrete subgroups Γ. In [16]the approach in [5] was extended using some results from the theory of algebraicgroups and arithmetic subgroups, and some new examples of Anosov automor-phisms were constructed. A large class of examples of Anosov automorphismswere constructed in [31] by associating nilpotent Lie groups to graphs and study-ing their automorphism group in relation to the graph. Study of the automor-phism groups of nilpotent Lie groups has also been applied to construct examplesof nilmanifolds that can not admit Anosov automorphisms (see [16] and [26]).Subsequently Anosov automorphisms of nilmanifolds have been constructedvia other approaches; the reader is referred to [46] and [40] and various referencescited there for further details. There has however been no characterisation ofnilmanifolds admitting Anosov automorphisms, and it may be hoped that furtherstudy of the automorphism groups of nilpotent Lie groups may throw more lighton the issue. 27 .3 Distal actions
An action of a group H on a topological space X is said to be distal if for any pairof distinct points x, y in X the closure of the H -orbit { ( gx, gy ) | g ∈ G } of ( x, y )under the componentwise action of G on the cartesian product space X × X doesnot contain a point on the “diagonal”, namely of the form ( z, z ) for any z ∈ X .The action on a locally compact group G by a group H of automorphisms of G isdistal if and only if under the H -action on G the closure the orbit of any nontrivialelement g in G does not contain the identity element.Distality of actions is a classical topic, initiated by D. Hilbert, but the earlystudies were limited to actions on compact spaces. The question of distalityof actions on R n , n ≥
1, by groups of linear transformations was initiated byC. C. Moore [69] and was strengthened by Conze and Guivarc’h [13]; see alsoH. Abels [1], where the results are extended to actions by affine transformations.It is proved that the action of a group H of linear automorphisms of R n is distalif and only if the action of each h ∈ H (viz. of the cyclic subgroup generated byit) is distal, and that it holds if and only if all (possibly complex) eigenvalues of h are of absolute value 1 (see [13], [1]).It is proved in [2] that the action of a group H of automorphisms of a connectedLie group G is distal if and only if the associated action of H on the Lie algebra G of G is distal (to which the above characterisations would apply).For a subgroup H of GL( n, R ), the H -action on R n is distal if and only ifthe action of its algebraic hull (Zariski closure) in GL( n, R ) is distal, and theaction of an algebraic subgroup H of GL( n, R ) on R n is distal if and only if theunipotent elements in H form a closed subgroup U (which would necessarily bethe unipotent radical) such that H/U is compact. (see [1], Corollaries 2.3 and 2.5).Via the above correspondence these results can be applied also to actions of groupsof automorphisms of a general connected Lie group G .An action of a group H on a space X is called MOC (short for “minimal orbitclosure”) if the closures of all orbits are minimal sets (viz. contain no propernonempty invariant closed subsets). When X is compact, the MOC condition isequivalent to distality. For the action on a topological group G by a group ofautomorphisms MOC implies distality. The converse is known in various cases,including for actions on connected Lie groups G by groups of automorphisms (see[2]). The reader is also referred to [3], [80] and [78] for some generalisations of thisas well as some of the other properties discussed above to more general locallycompact groups).A connected Lie group G is said to be of type R if for all g ∈ G all (possiblycomplex) eigenvalues of Ad g are of absolute value 1. Thus in the light of theresults noted above G is of type R if and only if the action of Inn( G ) on G is28istal. This condition is also equivalent to G having polynomial growth, viz. forany compact neighbourhood V of the identity the Haar measure of V n grows atmost polynomially in n ; see [60] and also [49] and [68] for related general results. A homeomorphism ϕ of a compact metric space ( X, d ) is said to be expansive ifthere exists ǫ > x, y there exists aninteger n such that d ( ϕ n ( x ) , ϕ n ( y )) > ǫ ; the notion may also be defined in termsof a uniformity in place of a metric. An automorphism τ of a topological group G is expansive if there exists a neighbourhood V of the identity such that for anynontrivial element g in G there exists an integer n such that τ n ( g ) / ∈ V . Moregenerally the action of a group Γ of automorphisms of a topological group G issaid to be expansive if there exists a neighbourhood V of the identity in G suchthat for any nontrivial g in G there exists a γ ∈ Γ such that γ ( g ) / ∈ V .A compact connected topological group admits a group of automorphisms act-ing expansively only if it is abelian (cf. [65]) and finite dimensional (cf. [66]). Forcompact abelian groups the expansiveness condition on actions of automorphismgroups has been extensively studied using techniques of commutative algebra (see[79] for details).If G is a connected Lie group with Lie algebra G , the action of a subgroup Γof Aut( G ) on G is expansive if and only if for the induced action of Γ on G , forany nonzero ξ ∈ G the Γ-orbit of ξ is unbounded in G (cf. [9], where the issue isalso considered for semigroups of endomorphisms). It is also deduced in [9] thatif Γ is a virtually nilpotent Lie group (viz. with a nilpotent subgroup of finiteindex) of Aut( G ) whose action on G is expansive, then Γ contains an elementacting expansively on G . The action of Aut( G ) on G , where G is a Lie group, induces an action of Aut( G )on the space of probability measures on G (see below for details). This actionplays an important role in many contexts. This section is devoted to recallingvarious results about the action and their applications. Let G be a connected Lie group and as before let P ( G ) denote the space ofprobability measures on G . We consider P ( G ) equipped with the weak ∗ topology29ith respect to the space of bounded continuous functions; we note that thetopology is metrizable, and a sequence { µ j } in P ( G ) converges to µ ∈ P ( G ) ifand only if R G f dµ j → R G f dµ , as j → ∞ , for all bounded continuous functions f on G . We recall that the action of Aut( G ) on G induces an action on P ( G ),defined, for τ ∈ Aut( G ) and µ ∈ P ( G ), by τ ( µ )( E ) = µ ( τ − ( E )) for all Borelsubsets E of G .For λ, µ ∈ P ( G ), λ ∗ µ denotes the convolution product of λ and µ , and for any µ ∈ P ( G ) and k ∈ N we denote by µ k the k -fold convolution product µ ∗ · · · ∗ µ .Behaviour of orbits of probability measures under actions of various subgroupsis of considerable interest in various contexts. One of the issues is to understandconditions under which orbits of automorphism groups A ⊂ Aut( G ) are locallyclosed, namely open in their closures. We recall here that by a result of Effros [42]this condition is equivalent to a variety of other “smoothness” conditions for theaction, including that the orbits map being an open (quotient) map onto its image(the latter being considered with respect to the induced topology from P ( G )).As before for a connected Lie group G we view Aut( G ) as a subgroup of GL( G ),where G is the Lie algebra of G , and a subgroup A of Aut( G ) is said to be almostalgebraic if it is an almost algebraic subgroup of GL( G ) (see § Theorem 6.1. ( cf. [36], Theorem 3.3 ) Let G be a connected Lie group and A bean almost algebraic subgroup of Aut( G ) . Let C be the maximal compact subgroupcontained in the center of G . Suppose that for any g ∈ G , { g − τ ( g ) | τ ∈ A } ∩ C is finite. Then for any µ ∈ P ( G ) the A -orbit { τ ( µ ) | τ ∈ A } is open in its closurein P ( G ) . Moreover, if A consists of unipotent elements in GL( G ) then the A -orbitis closed in P ( G ) . The theorem implies in particular that when G has no compact subgroup ofpositive dimension contained in its center, for the action of any almost algebraicsubgroup A of Aut( G ) (which includes also the whole of Aut( G ) in this case) theorbits of A are locally closed. Let µ ∈ P ( G ). We denote by G ( µ ) the smallest closed subgroup of G containingsupp µ , the support of µ , by N ( µ ) the normaliser of G ( µ ) in G and by Z ( µ ) thecentraliser of supp µ , namely { g ∈ G | gx = xg for all x ∈ supp µ } .A λ ∈ P ( G ) is called a factor of µ if there exists ν ∈ P ( G ) such that µ = λ ∗ ν = ν ∗ λ . It is known that any factor of µ has its support contained in N ( µ ),more specifically in a coset of G ( µ ) contained in N ( µ ) (see [32], Proposition 1.1for the first assertion; the second is an easy consequence of the first).30 question of interest, which is not yet understood in full generality, is whethergiven µ ∈ P ( G ) and a sequence { λ j } of its factors there exists a sequence { z j } in Z ( µ ) such that { λ j z j } is relatively compact in P ( G ) (in place of Z ( µ ) onemay also allow in this respect the subgroup { g ∈ G | gµg − = µ } , which canbe bigger, but the distinction turns out to be a rather technical issue, and weshall not concern ourselves with it here). It is known that given a sequence offactors { λ j } there exists a sequence { x j } in N ( µ ) such that { x j µx − j } , { x − j µx j } and { x j λ j } are relatively compact [32], [33], and it suffices to show that for sucha sequence { x j } the sequence of cosets { x j Z ( µ ) } is relatively compact in G/Z ( µ ).It is known that this is true when G is an almost algebraic Lie group (and alsounder some weaker, somewhat technical, conditions) (see [32], [36]); the proofshowever fall back on reducing the question to vector space situation and a moredirect approach connecting the question to Theorem 6.1 would be desirable.A similar issue, or rather a more general one, arises in the study of the decayof concentration functions of µ n as n → ∞ , where one would like to know underwhat conditions a sequence of the form { x j µx − j } is relatively compact. We shallhowever not go into the details of the concept or the results about it here; atreatment of the topic in this perspective may be found in [21] and [39]; for acomplete resolution of the problem itself, which however involves a somewhatdifferent approach, the reader is referred to [59] and other references cited there. A locally compact group is called a
Tortrat group if for any µ ∈ P ( G ) which isnot idempotent (viz. such that µ = µ ) the closure of { gµg − | g ∈ G } in P ( G )does not contain any idempotent measure. We note that the issue concerns theclosure of the orbits of µ in P ( G ) under the action of the subgroup Inn( G ); thelatter being contained in Aut ( G ) which is an almost algebraic subgroup is usefulin this respect.Recall (see § R if for all g ∈ G all eigenvalues of Ad g are of absolute value 1. It was shown in [37] that fora Lie group the two properties are equivalent: Theorem 6.2. (cf. [37])
A connected Lie group is a Tortrat group if and only ifit is of type R . The result implies also that a connected locally compact group is a Tortratgroup if and only if it is of polynomial growth (see [37] for details and the relatedreferences). 31 .4 Convergence of types
Let { λ j } and { µ j } be two sequences in P ( G ) converging to λ and µ respectively.Suppose further that there exists a sequence { τ j } in Aut( G ) such that τ j ( λ j ) = µ j for all j . A question, arising in various contexts in the theory of probabilitymeasures on groups, is under what (further) conditions on λ and µ can we concludethat there exists a τ ∈ Aut( G ) such that τ ( λ ) = µ ; the reader can readily convinceherself/himself that further conditions are indeed called for. When there exists a τ as above we say that convergence of types holds. Though the term “type of ameasure” does not seem to make an appearance in literature freely, implicit in theterminology above is the idea that { τ ( µ ) | τ ∈ Aut( G ) } constitutes the “type of µ ”, or that µ and τ ( µ ) are of the same type for any τ ∈ Aut( G ), and the questionis if you have pair of sequences of measures with the corresponding measures ofthe same type, converging to a pair of measures, under what conditions can weconclude the limits to be of the same type.We now recall some results in this respect; for reasons of simplicity of exposi-tion we shall not strive for full generality (see [20] for more details). As in § G ) be the group of affine automorphisms, and let ρ : Aff( G ) → GL( V ) be thelinearising representation (see § ρ is defined over V = ∧ a B , where B is the Lie algebra of Aff( G ) and a is the dimension of Aut( G ).We say that µ ∈ P ( G ) is ρ -full if there does not exist any proper subspace U of V that is invariant under ρ ( g ) for all g ∈ supp µ ; though this condition is rathertechnical, as it involves the linearising representation, it is shown in [20] that forvarious classes of groups it holds under simpler conditions; for instance if G isan almost algebraic subgroup of GL( n, R ) then the condition holds for any µ forwhich supp µ is not contained in a proper almost algebraic subgroup of G . Thefollowing is a result which is midway between the rather technical Theorem 1.5and the specialised result in Theorem 1.6 from [20], whose proof can be read offfrom the proof of Theorem 1.6. Theorem 6.3.
Let G be a connected Lie group. Let { λ j } and { µ j } be sequencesin P ( G ) converging to λ and µ respectively. Suppose that there exists a sequence { τ j } in Aut ( G ) such that µ j = τ j ( λ j ) for all j and that λ and µ are ρ -full. Thenthere exist sequences { θ j } and { σ j } in Aut( G ) such that { θ j } is contained in acompact subset of Aut( G ) , { σ j } are isotropic shear automorphisms, and τ j = θ j σ j for all j . If moreover supp λ is not contained in any proper closed normal subgroup M of G such that G/M is a vector space of positive dimension, then { σ j } is alsorelatively compact. We note that when { τ j } as in the theorem is concluded to be relatively com-pact, convergence of types holds (for the given sequences); with the notation as32n the theorem, if τ is an accumulation point of { τ j } then τ ( λ ) = µ . In the lightof the conclusion of the theorem it remains mainly to understand the asymptoticbehaviour under sequences of shear automorphisms.A special case of interest is when { λ j } are all equal, say λ . For this case we re-call also the following result, for shear automorphisms, proved in [29] (Theorem 4.3there); the result played an important role in the proof of the main theorem thereconcerning embeddability in a continuous one-parameter semigroup, for a class ofinfinitely divisible probability measures; we shall not go into the details of theseconcepts here. It may be noted that one of the conditions in the hypothesis is asin statement (i) of Theorem 4.4. Theorem 6.4.
Let G be a connected Lie group and T be a torus contained inthe center of G . Let H be a closed normal subgroup of G such that G/H istopologically isomorphic to R n for some n ≥ . Let W = G/H and θ : G → W be the canonical quotient homomorphism. Let λ ∈ P ( G ) be such that θ ( µ ) n isabsolutely continuous with respect to the Lebesgue measure on W . Let { ϕ j } bea sequence of homomorphisms of W into T and for all j let τ j ∈ Aut( G ) be theshear automorphism of G corresponding to ϕ j . Suppose that the sequence { τ j ( λ ) } converges to a measure of the form τ ( λ ) for some τ ∈ Aut( G ) fixing T pointwise.Then there exists a sequence { σ j } in Aut( G ) such that σ j ( µ ) = µ and { τ j σ j } isrelatively compact. References [1] Herbert Abels, Distal affine transformation groups, J. Reine Angew. Math. 299/300(1978), 294 - 300.[2] Herbert Abels, Distal automorphism groups of Lie groups, J. Reine Angew. Math.329 (1981), 82 - 87.[3] Herbert Abels, Which groups act distally? Ergodic Theory Dynam. Systems 3(1983), no. 2, 167 - 185.[4] Nobuo Aoki, Dense orbits of automorphisms and compactness of groups, TopologyAppl. 20 (1985), no. 1, 1 - 15.[5] Louis Auslander and John Scheuneman, On certain automorphisms of nilpotentLie groups, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley,Calif., 1968) pp. 9 - 15, Amer. Math. Soc., Providence, R.I., USA.[6] M. Bachir Bekka and Matthias Mayer, Ergodic theory and topological dynamics ofgroup actions on homogeneous spaces, London Mathematical Society Lecture NoteSeries, 269. Cambridge University Press, Cambridge, 2000. x+200 pp.
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S.G. DaniDepartment of MathematicsIndian Institute of Technology BombayPowai, Mumbai 400005IndiaE-mail: [email protected]@math.iitb.ac.in