Locally compact transformation groups

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

This paper is concerned with the topological structure of solvmanifolds i.e. spaces on which solvable Lie groups of transformations act transitively. The special case of nilmanifolds in which the group of transformations is nilpotent has been studied by Malcev [5]. Examples of two dimensional solvmanifolds are the torus, Klein bottle, cylinder, mobius strip. The central results are THEOREM A. Two compact solvmanifolds having the same fundamental group are isomorphic. THEOREM B. Any solvmanifold is covered a finite number of times by the direct product of a compact solvmanifold and a euclidean space; moreover, the covering group is abelian. THEOREM C. If S is an algebraically subgroup (cf. ?9 for definition) of the solvable Lie group G, then G/S is homeomorphic to the direct product of a compact solvmanifold and a euclidean space. Theorem C applies for example, if S is a closed subgroup which is generated by its connected identity-component together with elements s1, s2, -such that all the eigenvalues of Adsi are positive (i = 1, 2, * ). The assertion in Theorem A, applied to nilmanifolds, was proved by Malcev [5]. The situation in that case is simplified by virtue of the fact that two nilpotent groups which operate transitively and effectively on spaces with same fundamental group are isomorphic-a fact which is not true in the case of general solvmanifolds. Accordingly, there is a natural homeomorphism between nilmanifolds with isomorphic fundamental groups but there is no natural homeomorphism in the case of solvmanifolds. In the absence of uniqueness, therefore, proof of the existence of the homeomorphisin in Theorem A is rather elusive. After a number of algebraic maneuvers, proof of Theorem A is reduced to comparing two fibre bundles over homeomorphic toroids-the fibres being the orbits of transformation groups which are isomorphic as abstract topological groups. The main topological lemma asserts the existence of an equivariant homeomorphism between the two bundles. The major algebraic fact which underlies our method is the theorem of ?5 on uniform subgroups of solvable groups. In Theorem B, the covering need not be univalent-as is illustrated by the mobius band. The sufficiency condition in Theorem C yields as a special case a result of Chevalley: the factor space of a solvable Lie group by a connected subgroup is a product of circles and straight lines. Theorem A yields more generally

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting — for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Janich 1968), (Palais 1960), (Schultz 1984).

Let X be a uniform space, let T be a multiplicative topological group, and let T act as a transformation group on X. A subset A of T is said to be (left) syndetic provided that T= AK for some compact subset K of T. T he transformation group T is said to be almost periodic on X provided that if a is an index of X, then there exists a syndetic subset A of T such that xA Cxa for all xEX. If xGX, then the transformation group T is said to be locally almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x and a syndetic subset A of T such that VA CU. The transformation group T is said to be locally almost periodic on X in case T is locally almost periodic at every point of X. If xEX, then the transformation group T is said to be locally weakly almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x, a syndetic subset A of T, and a compact subset C of T such that y E V implies the existence of a subset B of T for which A CBC and yBC U. It is readily proved that if xcX, then T is locally weakly almost periodic at x if and only if for each neighborhood U of x there exist a neighborhood V of x and a compact subset K of T such that VTC UK. If xEX, then the transformation group T is said to be equicontinuous at x provided that if a is an index of X, then there exists an index e of X such that xftCxta for all tC T. The transformation group T is said to be equicontinuous on X in case T is equicontinuous at every point of X. The transformation group T is said to be uniformly equicontinuous on X provided that if a is an index of X, then there exists an index ,B of X such that xl3tCxta for all xEX and all tcT. It is readily verified that if X is compact, then T is uniformly equicontinuous if and only if T is equicontinuous. The transformation group T is said to be distal on X provided that if x, yEX with x5-zy, then there exists an index a of X such that (xt, yt) Ea for all t E T. We also consider T to be a transformation group acting on X X X in the following manner: if x, yEX and if tET, then (x, y)t is defined to be (xt, yt).

In studying the actions of a compact group on a topological space X, one very soon arrives at the conclusion that the smaller the codimension of the largest orbit of the action the more manageable the problem becomes. For example, if G acts on euclidean n-space with an orbit of dimension greater than n 3, then G essentially acts linearly [5]. Similarly if G is a simply connected Lie group acting on itself transitively, then the action is essentially left translation. Unfortunately, this at least demands that the dimension of G itself be relatively large with respect to that of X. However, these results suggest that in order to study the action of an arbitrary compact group H on X, we might attempt to imbed this action in an action of a larger compact group G on X hence conceivably raising the dimension of the orbits. In this paper we will be concerned with the following type of problem. If H is an invariant subgroup of G' and we are given an action of G' on X, then we have associated canonically an action of G'/H on the orbit space X/H [6, p. 61]. When is the converse true? That is, given H acting on X and G'/H acting on the orbit space X/H when can we obtain a consistent action of G' on X? We will restrict ourselves throughout to the case where H acts freely on X and G' = G x H, though many of the theorems can be proved in the more general setting of [8]. With H acting freely, we arrive at the situation of a principal fibre space in the sense of Cartan. In ? 2 we study the case where G = G'/H acts transitively on the orbit space B = X/H which is then the base space of this fibre space.

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

This chapter consists of a concise exposition of the structure and classification theories of compact Lie groups and their representations from the geometric viewpoint of transformation groups. An explicit and neat understanding of the orbit structure of the adjoint action of a compact Lie group G plays a central role in the classification theory developed by E. Cartan and H. Weyl. This more geometric approach is actually more natural and straightforward than the usual Lie-algebra-theoretical approach. Furthermore, such an approach will also provide us with valuable examples and insight for later investigation of topological transformation groups.

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

We investigate the representation theory of the crossed–product C*–algebra associated with a compact group G acting on a locally compact space X when the stability subgroups vary discontinuously. Our main result applies when G has a principal stability subgroup or X is locally of finite G–orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation V of a stability subgroup is obtained by restricting V to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of V. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup; the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the C*–algebra of the motion group ℝn ⋊ SO(n) is a Fell algebra. This uses the classical branching theorem for the special orthogonal group SO(n) with respect to SO(n − 1). Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Covariant and equivariant formality theorems

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

Let S be a closed surface of genus g greater than zero. In the present paper, we study the topological-dynamical action of the mapping class group on the \mathbb T^n -character variety giving necessary and sufficient conditions for \mathrm{Mod}(S) -orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's theorem concerning inhomogeneous Diophantine approximation.

Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups.In §1, we show that many of the interesting properties of the proximal relation can be characterized in terms of a group subset that arises naturally from the generating subgroup.A general criterion for distal in terms of subnormality of the generating subgroup is shown in §2.Proofs are given for [9, Remark (2.1) and Theorem (2.2)], establishing that coset transformation groups of nilpotent groups are distal.§3 is concerned with distal in relation to certain coset transformation groups that arise from solvable Lie groups.Finally, we establish in §4 some results concerning the existence of subgroups which yield either point-transitive or minimal distal actions on the phase space.These results are strengthened when applied to certain discrete flows and examined in Lie groups.Proofs for the rest of the results of [9] are given in this section.Let G be a topological group and H a subgroup of G. Consider the space H\G={Hg | ge G} of right cosets of G.The coset transformation group of G induced by H is defined to be the point-to-point transitive transformation group (H\G, G) with action (Hfi g) -> Hfg.The fact that (H\G, G) is indeed a transformation group under this action follows from [6, Definition 1.39].If H is syndetic in G, H\G is compact uniformizable, whence H\G has a unique compatible uniformity.Thus, the proximal relation of (H\G, G) is well defined and will be denoted by P(H\G).Throughout this paper, G will denote a topological group with identity e and 77 will denote a syndetic subgroup.If K and L are subgroups of G, we denote that K is normal in L by K <] L. If K is a subgroup of G, then K* will denote the closure of K.The additive groups of integers and reals will be denoted by Z and R respectively.The neighborhood filter of e will be denoted by r¡e.For other definitions, see [9].Finally, the author would like to thank Professor W. Gottschalk for his stimulating comments concerning the results in this paper.2. The proximal relation.When the phase space is not necessarily uniform, we can still introduce a notion of proximal.

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...]