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For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 2 is an even integer, we show the following. If there exists a map between Lp(LG) and an ultraproduct of Lp(M) \otimes S_p(L^2 G) that intertwines the Fourier multiplier with the Schur multiplier, then G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups., Trans. Amer. Math. Soc., to appear

We study properly discontinuous and cocompact actions of a discrete subgroup Γ \Gamma of an algebraic group G G on a contractible algebraic manifold X X . We suppose that this action comes from an algebraic action of G G on X X such that a maximal reductive subgroup of G G fixes a point. When the real rank of any simple subgroup of G G is at most one or the dimension of X X is at most three, we show that Γ \Gamma is virtually polycyclic. When Γ \Gamma is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.

The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebra A is singly generated, which means that there exists an element x є A that is not contained in any proper sub-C*- algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group C*-algebras are singly generated. In particular, C*r (F ∞) ⨂ C*r (F ∞) is singly generated.

For a locally compact group G and 1 < p < ∞, let Ap(G) be the Figa-Talamanca-Herz algebra. Then the multiplier algebra MA p (G) of Ap(G) is a dual space. We say that Ap(G) has the approximation property (or simply, AP) in MA p (G) if there is a net {u α } in Ap(G) such that u α → 1 in the associated weak* topology. We prove that Ap(G) has the AP in MA p (G) if and only if there exists a net {a α } in Ap(G) such that ∥a α a - a∥ Ap (G) → 0 uniformly for a in any compact subset of Ap(G). Consequently, we have that if Ap(G) has the AP in MA p (G), then Ap(G) has the approximation property as a Banach space in the sense of Grothendieck for a discrete group G. We also study the relationship between the AP of Ap(G) in MA p (G) and the weak amenability of G.

For a ( d + 1 ) (d+1) -dimensional hyperbolic manifold M \mathcal {M} , we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group Γ \Gamma of M \mathcal {M} to be a discrete subgroup of S O e ( d + 1 , 1 ) SO_e(d+1, 1) with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl’s law, we obtained a lower estimate Ω ± ( x d / 2 ( log ⁡ log ⁡ x ) 1 / ( d + 1 ) log ⁡ x ) \Omega _{\pm }(\tfrac {x^{d/2}(\log \log x)^{1/(d+1)}}{\log x}) as x x goes to ∞ \infty .

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient E\L, where L is a connected and simply connected nilpotent Lie group and E is a torsion-free uniform discrete subgroup of L×C, with C a compact subgroup of Aut(L). We show that if the Lie algebra of L is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.

If X X is a proper C A T ( − 1 ) CAT\left ( -1\right ) -space and Γ \Gamma a non-elementary discrete group of isometries acting properly discontinuously on X , X, it is shown that the geodesic flow on the quotient space Y = X / Γ Y=X/\Gamma is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂ X \partial X and the non-wandering set of the flow equals the whole quotient space of geodesics G Y := G X / Γ GY:=GX/\,\Gamma (the latter being redundant when Y Y is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete C A T ( − 1 ) CAT\left ( -1\right ) -spaces by a one-ended group of isometries and (C) finite n n -dimensional ideal polyhedra.

In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our theory is valid for any (possibly infinite) discrete group. Some applications to topology are also given.

Let Γ \Gamma be a geometrically finite discrete group of isometries of hyperbolic space H F n \mathcal {H}_{\mathbb {F}}^n , where F = R , C , H \mathbb {F}= \mathbb {R}, \mathbb {C}, \mathbb {H} or O \mathbb {O} (in which case n = 2 n=2 ). We prove that the critical exponent of Γ \Gamma equals the Hausdorff dimension of the limit sets Λ ( Γ ) \Lambda (\Gamma ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group r with an irreducible unitary representation p and satisfying u(yz) = p(Qy)u(z). The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, R_ and R+, for the solution operator. The scattering operator, which maps R f into R+f, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of p is one, the elements of the scattering operator cannot vanish. However when dim(p) > 1 this is no longer the case.

Let G G be a σ \sigma -compact infinite locally compact group, and let L I M LIM be the set of left invariant means on L ∞ ( G ) {L^\infty }(G) . We prove in this paper that if G G is amenable as a discrete group, then L I M LIM has no exposed points. We also give another proof of the Granirer theorem that the set L I M ( X , G ) LIM(X,G) of G G -invariant means on L ∞ ( X , β , p ) {L^\infty }(X,\beta ,p) has no exposed points, where G G is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space ( X , β , p ) (X,\beta ,p) .

Let G be a locally compact group, let C,' (G) (resp. VN(G)) be the C*-algebra (resp. the von Neumann algebra) associated with the left regular representation I of G, let A(G) be the Fourier algebra of G, and let MOA(G) be the set of completely bounded multipliers of A(G) . With the completely bounded norm, MOA(G) is a dual space, and we say that G has the approximation property (AP) if there is a net {ua} of functions in A(G) (with compact support) such that ua -a 1 in the associated weak *-topology. In particular, G has the AP if G is weakly amenable (< A(G) has an approximate identity that is bounded in the completely bounded norm). For a discrete group F, we show that F has the AP X C,' (F) has the slice map property for subspaces of any C*-algebra X VN(JT) has the slice map property for a-weakly closed subspaces of any von Neumann algebra (Property Se) . The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group extensions. We also obtain some results concerning crossed products. For example, we show that the crossed product M ?a G of a von Neumann algebra M with Property S, by a group G with the AP also has Property S, .

The author generalizes B. Harris’ definition of harmonic volume to the algebraic cycle W k − W k − {W_k} - W_k^- for k > 1 k > 1 in the Jacobian of a nonsingular algebraic curve X X . We define harmonic volume, determine its domain, and show that it is related to the image ν \nu of W k − W k − {W_k} - W_k^- in the Griffiths intermediate Jacobian. We derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the k k -fold symmetric product of X X . We show that ν \nu , when applied to a certain class of forms, takes values in a discrete subgroup of R / Z {\mathbf {R}}/{\mathbf {Z}} and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if k ≥ 2 k \geq 2 . This implies that the image of W k − W k − {W_k} - W_k^- is identically zero in the Griffiths intermediate Jacobian if k ≥ 2 k \geq 2 . We introduce a new type of intermediate Jacobian which, like the Griffiths intermediate Jacobian, varies holomorphically with moduli, and we consider a holomorphic torus bundle on Torelli space with this fiber. We use the relationship mentioned above between ν \nu and harmonic volume to compute the variation of ν \nu when considered as a section of this bundle. This variational formula allows us to show that the image of W k − W k − {W_k} - W_k^- in this intermediate Jacobian is nondegenerate.

This paper deals with the p p -adic completion F p ∞ X {F_{p\infty }}X developed by Bousfield-Kan for a space X X and prime p p . A space X X is called F p {F_p} -good when the map X → F p ∞ X X \to {F_{p\infty }}X is a mod - p \bmod \text {-}p homology equivalence, and called F p {F_p} -bad otherwise. General examples of F p {F_p} -good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be F p {F_p} -bad. This provides the first example of an F p {F_p} -bad space of finite type and implies that the p p -profinite completion of a free group on two generators must have nontrivial higher mod - p \bmod \text {-}p homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the p p -adic completion can be extended to the wider class of p p -seminilpotent spaces.

In this paper we study the relationship between amenability, inner amenability and property P P of a von Neumann algebra. We give necessary conditions on a locally compact group G G to have an inner invariant mean m m such that m ( V ) = 0 m(V) = 0 for some compact neighborhood V V of G G invariant under the inner automorphisms. We also give a sufficient condition on G G (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan’s property T T , e.g., SL ( n , Z ) {\text {SL}}(n,\mathbb {Z}) , n ≥ 3 n \geq 3 ) such that each linear form on L 2 ( G ) {L^2}(G) which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach G G -modules is also obtained.

A locally compact topological group G is called an (H) group if G has a maximal compact normal subgroup with Lie factor. In this note, we study the problem when a locally compact group is an (H) group. Let G be a locally compact Hausdorff topological group. Let Go be the identity component of G. If G/Go is compact, then we say that G is almost connected. The structure of almost connected locally compact groups is well understood and it is the consequence of the solution of the Hilbert fifth problem. Roughly speaking, the structure of almost connected locally compact groups can be described by the following statement: Every almost connected locally compact group has small normal subgroups such that the quotient groups are Lie groups and the group itself is the inverse limit of these Lie groups in a natural way. We cannot expect locally compact groups in general to have such properties, so we ask when a locally compact group has a compact normal subgroup such that the quotient group is a Lie group. The purpose of the present note is to provide some partial answers to this question. Now, let G be a locally compact group. We may view G as an extension of Go by a totally disconnected locally compact topological group G/Go. Since the natures of these two types of groups are very different, it is convenient to study them separately first and then put results together afterwards. Because our knowledge of connected locally compact groups is by far superior to what we know about totally disconnected locally compact groups, our main effort lies on totally disconnected locally compact groups. Actually, we do not lose any generality approaching our problem by this method. If G/Go has a compact normal subgroup such that the quotient group is a Lie group (i.e., a discrete group), then G has an open normal subgroup G, such that G1 /Go is compact. Then G1 has a maximal compact normal subgroup K so that G1 /K is a Lie group, since G, is almost connected. Since K is characteristic in G1, it is normal in G. Therefore G/K is a Lie group. Notice that every totally disconnected locally compact group has small compact open subgroups. Frequently, Received by the editors July 26, 1988 and, in revised form, May 3, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22A05. ( 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

A subset E E of an infinite discrete group G G is called (i) an R W {R_W} -set if any bounded function on G G supported by E E is weakly almost periodic, (ii) a weak p p -Sidon set ( 1 ≤ p > 2 ) (1 \leq p > 2) if on l 1 ( E ) {l^1}(E) the l p {l^p} -norm is bounded by a constant times the maximal C ∗ {C^*} -norm of l 1 ( G ) {l^1}(G) , (iii) a T T -set if x E ∩ E xE \cap E and E x ∩ E Ex \cap E are finite whenever x ≠ e x \ne e , and (iv) an F T FT -set if it is a finite union of T T -sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group G G contains an R W {R_W} -set which is not an F T FT -set; (b) countable weak p p -Sidon sets, 1 ≤ p > 4 / 3 1 \leq p > 4/3 are F T FT -sets.

Let G G be a locally compact group, A ( G ) A(G) the Fourier algebra of G G , B ( G ) B(G) the Fourier-Stieltjes algebra of G G and VN ( G ) {\text {VN}}(G) the von Neumann algebra generated by the left regular representation λ \lambda of G G . Then A ( G ) A(G) is the predual of VN ( G ) {\text {VN}}(G) ; VN ( G ) {\text {VN}}(G) is a B ( G ) B(G) -module and A ( G ) A(G) is a closed ideal of B ( G ) B(G) . Let AP ( G ^ ) = { T ∈ VN ( G ) : u ↦ u ⋅ T {\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T is a compact operator from A ( G ) A(G) into VN ( G ) } {\text {VN}}(G)\} , the space of almost periodic operators in VN ( G ) {\text {VN}}(G) . Let C δ ∗ ( G ) C_\delta ^*(G) be the C ∗ {C^*} -algebra generated by { λ ( x ) : x ∈ G } \{ \lambda (x):x \in G\} . Then C δ ∗ ( G ) ⊂ AP ( G ^ ) C_\delta ^*(G) \subset {\text {AP}}(\hat G) . For a compact G G , let E E be the rank one operator on L 2 ( G ) {L^2}(G) that sends h ∈ L 2 ( G ) h \in {L^2}(G) to the constant function ∫ h ( x ) d x \int {h(x)dx} . We have the following results: (1) There exists a compact group G G such that E ∈ AP ( G ^ ) ∖ C δ ∗ ( G ) E \in \text {AP}(\hat G)\backslash C_\delta ^*(G) . (2) For a compact Lie group G G , E ∈ AP( G ^ ) ⇔ E ∈ C δ ∗ ( G ) ⇔ L ∞ ( G ) E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G) has a unique left invariant mean ⇔ G \Leftrightarrow G is semisimple. (3) If G G is an extension of a locally compact abelian group by an amenable discrete group then AP ( G ^ ) = C δ ∗ ( G ) {\text {AP}}(\hat G) = C_\delta ^*(G) . (4) Let G = F r G = {{\mathbf {F}}_r} , the free group with r r generators, 1 > r > ∞ 1 > r > \infty . If T ∈ VN ( G ) T \in {\text {VN}}(G) and u ↦ u ⋅ T u \mapsto u \cdot T is a compact operator from B ( G ) B(G) into VN ( G ) {\text {VN}}(G) then T ∈ C δ ∗ ( G ) T \in C_\delta ^*(G) .

We study the classification problem of crossed-product C ∗ {C^ * } -algebras of the form C r ∗ ( G ) × α χ Z C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf {Z}} , where G G is a discrete group, χ \chi is a one-dimensional character of G G , and α χ {\alpha _\chi } is the unique ∗ * -automorphism of C r ∗ ( G ) C_r^ * (G) such that if U U is the left regular representation of G G , then α χ ( U g ) = χ ( g ) U g {\alpha _{\chi }(U_{g})=\chi (g)U_{g}} , g ∈ G g \in G . When G = F n {G = F_{n}} , the free group on n n generators, we have a complete classification of these crossed products up to ∗ * -isomorphism which generalizes the classification of irrational and rational rotation C ∗ {C^ * } -algebras. We show that these crossed products are determined by two K K -theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural Aut ⁡ ( F n ) \operatorname {Aut} ({F_n}) -action, and that these two orbit invariants completely classify the characters up to automorphisms of F n {F_n} . The classification of crossed products follows from these results. We also consider the same problem for G G some other groups.

In this article we consider the Picard group SL(2, Z[i]) , viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold. INTRODUCTION It is well known that there is a close connection between the geodesic flow for the Modular surface and the continued fraction transformation on the unit interval. The relationship between the two is a consequence of the fact that the Fuchsian group associated to the Modular surface is the Modular group SL(2, Z) and the action of this group on the real line (as the "boundary" of the Poincare upper half plane) induces the continued fraction transformation on [0, 1] in a very natural way. Modern accounts of this theory can be found in the articles of Adler-Flatto [AF] and Series [S]. In this note we shall describe an analogous correspondence in which we replace the Modular Fuchsian group by the Picard Kleinian group SL(2, Z[i]) acting on the three-dimensional upper half plane model for hyperbolic space H3. In particular, we shall describe a transformation on the rectangle B = {z E C: I Re(z)I < 1/2, 0 < Im(z) < 1/2} which plays the role of the continued fraction transformation. As our main application of these ideas, we shall give an alternative approach to the meromorphic extension of the associated Zeta function, following the derivation for the case of the Modular surface described in the work of Mayer [M]. The construction we describe should have applications to the construction of special flows and also to the problem of the Markov spectrum of the Picard group (cf. Schmidt [Sc]). We hope to return to these problems later. 1. THE PICARD GROUP We define the Picard group F = SL(2, Z[i]) to be the group of 2 x 2 matrices whose entries are the Gaussian integers u + iv (u, v E Z) and such that every matrix has determinant 1 . Received by the editors September 30, 1993. 1991 Mathematics Subject Classification. Primary 30F40; Secondary 58F20. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

In response to one of ten problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretation— in the sense of Langlands—to the multiplicty in L 2 ( Γ ∖ G ) {L^2}(\Gamma \backslash G) of an irreducible unitary representation π \pi of a semisimple Lie group G G , where Γ \Gamma is a discrete subgroup of G G , in the case when π \pi has a highest weight.

With any topological group ( G , U ) (G, \mathcal {U}) one can associate a locally arcwise-connected group ( G , U ∗ ) (G, {\mathcal {U}}^{\ast }) , where U ∗ {\mathcal {U}}^{\ast } is stronger than U \mathcal {U} . ( G , U ) (G, \mathcal {U}) is a weakened Lie ( W L ) (WL) group if ( G , U ∗ ) (G, {\mathcal {U}}^{\ast }) is a Lie group. In this paper the author shows that the WL groups with which a given connected Lie group ( L , J ) (L,\mathcal {J}) is associated are completely determined by a certain abelian subgroup H H of L L which is called decisive. If L L has closed adjoint image, then H H is the center Z ( L ) Z(L) of L L ; otherwise, H H is the product of a vector group V V and a group J J that contains Z ( L ) Z(L) . J / Z ( L ) J/Z(L) is finite (trivial if L L is solvable). We also discuss the connection between these theorems and recent results of Goto.

Let G be a noncompact nondiscrete σ \sigma -compact locally compact metric group. A Baire category argument gives measurable sets { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} of finite measure with card ( Γ ) = c (\Gamma ) = c which are independent on the open sets. One approximates { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions { f γ : γ ∈ Γ } \{ {f_\gamma }:\gamma \in \Gamma \} in CB ( G ) {\text {CB}}(G) with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on CB ( G ) {\text {CB}}(G) there are 2 c {2^c} mutually singular elements of LIM each of which is singular to TLIM.

It is shown that for a discrete group of hyperbolic motions of the unit ball of R n {\mathbf {R}^n} , there is a single construction of fundamental regions which gives the Ford and Dirichlet regions as special cases and which also yields fundamental regions based at limit points. It is shown how the region varies continuously with the construction. The construction is connected with a class of limit points called Garnett points. The size of the set of such points is investigated.

1. Introduction. This paper contains proofs of the results announced in [5] concerning nonisomorphic pure subgroups of primary groups with prescribed socles. In fact, in the present paper, we strengthen and generalize the main theorenm of [5]. In addition, we establish the existence of a surprising abundance of essentially indecomposable groups and give a rather simple and general construction of infinite reduced primary groups without proper isomorphic subgroups. Some consideration is also given to those primary groups having the property that all their subsocles support pure subgroups. Although our most important results relate to primary groups with countable basic subgroups, they can clearly be generalized to groups having No as their critical number. A less trivial generalization to primary groups having greater cardinality than their basic subgroups is mentioned in our final section. All in all, this paper might be considered as one further contribution toward demonstrating the complexity of the structure of uncountable primary groups; for example, our Theorem 2.7 constitutes the most striking evidence known to the authors of the inadequacy of the Ulm factors of an uncountable primary group in determining the group. All groups are assumed to be primary abelian groups for some fixed prime p, and all topological references are to the p-adic topology. We follow the notation and terminology of [3]. The cardinal 2KO is denoted by c. We shall call S a subsocle of the primary group G if S is a subgroup of G[p]. A subsocle S supports a subgroup H if Hjp] -S. A subsocle S of G will be said to be a discrete subsocle if S flp'G = 0 for some positive integer n and a dense subsocle if S is dense in G[p] (relative to the p-adic topology of G). A noteworthy property of dense subsocles is the following: if S is a dense subsocle of G and H is a neat subgroup of G supported by S, then H is a pure subgroup of G (see [6]). Finally, recall that a group is said to be essentially indecomposable if every direct decomposition of the group involves a bounded summand.

enemy. Some of the theorems in the first two sections are modeled on those of Snapper, in [18], who considered these problems for everything discrete, the powerful techniques of homological algebra being available in this case. Unfortunately, there is no machinery of projective or injective resolutions immediately at our disposal for the cohomology theories based on Borel cochains. There is a type of resolution, which leans toward the injective, but its full homological character is not understood as yet. The entire idea of considering the cohomology of general group actions was inspired by the paper of Snapper. These things have been considered in the continuous case by Mostow in [17], where equivariant cohomology groups are defined and studied. Almost nothing being known about the higher cohomology groups, it was desirable to ferret out those isolated pockets of triviality. This is done in the last section, where it is shown that for G abelian locally compact separable metrizable, R the real line, G operating trivially on R, Hn(G, R), the bounded Borel groups defined in [16], do not depend upon the topology of G, nor its torsion. In fact, R seems to ignore everything about G except a certain torsion-free discrete group D associated to G. The further fact that these groups are isomorphic to the group of alternating multilinear functions An (G, R) was suggested by the results of Kleppner on H2(G, T), where G is locally compact abelian and T the circle group. He showed that for all G except a certain intractable class of 2-primary groups, whose fate is yet undecided, that H2(G, T)=L2(G, T)/S2(G, T), the continuous bihomomorphisms modulo the symmetric ones. We have not resolved this open question

In his paper, G. Shimura quotes a proposition [2, Proposition 2.5] which is derived directly from a transformation formula of theta functions due to KrazerPrym and suggests proving this proposition by means of a generalized Poisson summation formula obtained by A. Weil [4, Theoreme 4]. The purpose of this paper is to execute his idea. Let us explain briefly the result. Let V be a 2n dimensional real vector space and let D be a discrete subgroup of rank 2n in V. Suppose that we have a nondegenerate alternate bilinear form A on V x V which assumes integral values on D x D. The form A represents an integral cohomology class on the torus T= VID. A complex vector space structure J on V induces a complex structure on the torus, which may be denoted by the same notation. Such a complex structure J is said to be admissible if there is a positive divisor on the complex torus (T, J) whose cohomology class is A. Making use of the theory of theta functions on the complex vector space (V, J), we assign to each admissible complex structure J on T a holomorphic map 0(J) of the complex torus (T, J) into a complex projective space P of a certain dimension depending only on the form A, in such a way that the cohomology class A is a rational multiple of the image of the cohomology class of hyperplane sections on P under the cohomology homomorphism 0(J)*. Consider the group S(D) of real linear transformations of V which leave the bilinear form A and the subgroup D invariant. If a E S(D), then a induces an isomorphism a of T onto itself. If J is an admissible complex structure on T, there is a unique admissible complex structure Ja on T such that a: (T, Ja) -(T, J) is a holomorphic isomorphism of these two complex tori. Now, the result asserts that if we make a suitable choice of the assignment 0(J) to each admissible complex structure J and if a belongs to a congruence subgroup in S(D) of sufficiently high level with respect to an appropriate base of D, then the following diagram is commutative

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C S:C \to SC be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and H n ( X ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {F^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {f^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. This result is applied to show that if X is locally connected, then H 1 ( X ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 n > 1 . The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, H n ( X ) / Tor H n ( X ) {H^n}(X)/{\operatorname {Tor}}\;{H^n}(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that H n ( X ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.

motivating the study of such a problem. First, for an arbitrary locally compact semigroup, the closure of the maximal subgroup containing an idempotent forms a locally compact semigroup. If we know this maximal subgroup is a topological group, then it is a locally compact topological group. (This fact will be proved in the following paragraph.) By our rather good knowledge of locally compact topological groups, we can obtain information about the structure of this locally compact semigroup. Second, if we look at this problem from the point of view of transformation groups, we have a group G acting on a locally compact topological space. Naturally, we would like this group to be topologized so that it is a topological group. It is known that the weakest topology for G to be a topological group is the g-topology [1]. It is somewhat harder to deal with the g-topology than with the k-topology, but under some conditions the g-topology will coincide with the k-topology. Among the known conditions, the local compactness of the group seems most initeresting. If one tries to generalize such a condition, it is rather natural to ask the following: If G is a group of homeomorphisms of X onto X, and the closure of G is locally compact in the k-topology, does the induced topology of G coincide with the g-topology? Third, in the past few years there have been several papers about such questions: What is the relationship between the algebraic structure and the topological structure? When can one strengthen the topological structure so that even under weak conditions the algebraic operations will be continuous in a certain sense? For instance, in topological groups, if the group space is locally compact, and if the multiplication is separately continuous with respect to the topology of the space, then the group is a topological group. We are able to prove the same theorem when the space is second countable and second category without the assumption of local compactness [9]. Now, our present problem arises quite

Let G be an arbitrary lattice-ordered group. We define a topology on G, called the J \mathcal {J} -topology, which is a group and lattice topology for G and which is preserved by cardinal products. The J \mathcal {J} -topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the J \mathcal {J} -topology to be Hausdorff, and we investigate J \mathcal {J} -topology convergence.

For G a locally compact group and G ^ \hat G its dual, let M d ( G ^ ) {\mathcal {M}_d}(\hat G) be the C ∗ {C^ \ast } -algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on M d ( G ^ ) {\mathcal {M}_d}(\hat G) is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in M d ( G ^ ) {\mathcal {M}_d}(\hat G) are the discrete measures, and also the sup and lim sup norms are identical on M d ( G ^ ) {\mathcal {M}_d}(\hat G) . These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.

algebra; the group r becomes the boundary of the disc, and all functions take their maximum modulus on r. For interior points r of the disc, there is a measure on r such that the value at r is given by an integral of the boundary values. This "Poisson integral" is studied in detail, and it is shown that one of these "generalized holomorphic" functionsf cannot vanish on an open set of the boundary which has positive measure with respect to all of these harmonic measures. Moreover, when G has sufficiently many continuous "characters" (with values in the unit disc), and f vanishes on a nonvoid open set, then f =0. In particular, the algebra becomes an integral domain. We can thus make arbitrarily huge (semi-simple) Banach algebras which are integral domains. We wish to cite some work related to ours done by Goldman [VI] and by Mackey [VII]. Goldman states 3.1 when G is a totally ordered discrete group. Moreover, our 4.6 was motivated by a discussion with him concerning possible generalizations of the classical case of the disc. Mackey also states 3.1 and has a complicated Cauchy integral formula. His point of view is much

1. 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 I g e 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 (Hf, 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 H 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 Tie. For other definitions

0. Introduction. Let G denote an analytic group and let P c G denote a discrete subgroup. A left invariant Haar measure on G then induces a measure on G/r, which is determined up to a positive constant. We let v denote one such induced measure on G/P. DEFINITION 0.1. F is called a lattice, in case v(G/f) < oo. Unless specified otherwise, F will denote a lattice in the analytic group G. Let W be a simply connected Cr manifold; let M denote the set of all one-one homomorphisms

Let W be a UHF algebra of Glimm type n?, i.e., 9t 0 Nk, where N = N, = N2 = ... are n X n matrix algebras. We define an AF-subalgebra W G of 9t, consisting of those elements of 9. invariant under a group of automorphisms {ag: g E G = SU(n)} of product type. StG is shown to be generated by an embedding of S(oo), the discrete group of finite permutations on countably many symbols. Let co be a pure product state on 9t, xc its restriction to c G Let e E N be a one-dimensional projection with corresponding projections e k E N,. Then if both (i) kvlw(ek) = oo, and (ii) 0 < EklIw(ek)[w(ek)] < oo hold, WG is not pure. G is shown to be pure if there exist orthogonal one-dimensional projections {p,: I < i s n} of N with corresponding projections pk E Nk such that W(pk) = 0 or 1, I < i s n, k ? 1, and 0 < Ek2Iw(Pk) < 00 for at most one i.

For discrete groups we introduce a new class of sets, called weak uniqueness sets, which for abelian groups contains the class of sets of uniqueness. Considered is the problem of determining groups for which every finite set is a weak uniqueness set. Some examples are given.

A noncompact locally compact group G is called an Eberlein group if W(G) = B(G)- where W(G) is the algebra of continuous weakly almost periodic functions on G and B(G)- is the uniform closure of the Fourier-Stieltjes algebra of G. We show that if G is a noncompact (IN )-group or a noncompact nilpotent group then W(G)/B(G)- contains a linear isometric copy of l. In particular, G is not an Eberlein group. On the other hand, finite direct products of Eucidean motion groups and, by a result of W. Veech, noncompact semisimple analytic groups with finite centers are Eberlein groups. 1. Introduction. Let G be a locally compact group, C(G) the space of bounded continuous complex-valued functions on G with the sup norm, W(G) the space of continuous weakly almost periodic (w.a.p) functions on G, B(G) the Fourier-Stieltjes algebra of G and B(G)- its uniform closure in C(G). It is known that B(G)- C W(G) and if G is compact then C(G) = W(G) = B(G)-. Therefore, in this paper, G is assumed to be noncompact. In his efforts to characterize w.a.p. functions, W. F. Eberlein first raised the question whether, for abelian G, B(G)-= W(G). W. Rudin (24) made the initial contribution to this question by showing that B(G)- C W(G) if G is abelian and contains a discrete subgroup which is not of bounded order and D. Ramirez (23), by studying discrete abelian groups with bounded orders together with the structure theorem for locally compact abelian groups was able to show that Rudin's conclusion holds for all abelian groups, see the monographs (4 and 11) for complete accounts of this result. In this paper we will extend Rudin-Ramirez' result to include many nonabelian groups. Our main result is the following.

Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see ?1). In general, this topological structure of A (G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an infinite-dimensional vector space over an arbitrary field, then A (G) already fails to be locally compact. On the other hand, if G is a connected Lie group, we shall show without any difficulty that the compact-open topology of A (G) coincides with the topology obtained by identifying A (G) with a closed subgroup of the linear group of automorphisms of the Lie algebra of G, as was done by Chevalley (in [1]) in order to make A (G) into a Lie group. We shall then deduce that A (G) is a Lie group whenever the group of its components, G/Go, is finitely generated('), where Go denotes the component of the identity element in G. The other questions with which we shall be concerned are the following: Let I(Go) denote the group of the inner automorphisms of Go, and let E(Go, G) denote the natural image in A (Go) of A (G). Regard I(Go) and E(Go, G) as