Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

Hyperbolic groups have flat-rank at most 1

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.     

Equivariant sheaf cohomdlogy

In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)^G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology H_G(X;-).     

Linearization of proper group actions

We prove that if G is a locally compact group acting properly (in the sense of R. Palais) on a space X that is metrizable by a G-invariant metric, then X can be embedded equivariantly into a normed linear G-space E endowed with a linear isometric G-action which is proper on the complement E?{0}. If, in addition, G is a Lie group then E?{0} is a G-equivariant absolute extensor. One can make this equivariant embedding even closed, but in this case the non-proper part of the linearizing G-space E may be an entire subspace instead of {0}.     

On Infinitely Divisible Distributions on Locally Compact Abelian Groups

Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d.     

Orbit spaces and unions of equivariant absolute neighborhood extensors

Let G be a locally compact Hausdorff group. We study orbit spaces and unions of equivariant absolute neighborhood extensors (G-ANEs) in the category of all proper G-spaces that are metrizable by a G-invariant metric. We prove that if a proper G-space X is a G-ANE such that all the G-orbits in X are metrizable, then the G-orbit space X/G is an ANE. Equivariant versions of Hanner's theorem and Kodama's theorem about unions of absolute neighborhood extensors are established. We also introduce the notion of a G-polyhedron and prove that if G is any compact group, then every G-ANR is arbitrary closely dominated by a G-polyhedron. Each G-polyhedron is a G-ANE.     

Relative Kazhdan Property

We perform a systematic investigation of Kazhdan's relative Property (T) for pairs (G,X), where G is a locally compact group and X is any subset. When G is a connected Lie group or a p-adic algebraic group, we provide an explicit characterization of subsets X⊂G such that (G,X) has relative Property (T). In order to extend this characterization to lattices Γ⊂G, a notion of “resolutions” is introduced, and various characterizations of it are given. Special attention is paid to subgroups of SU(2,1) and SO(4,1).     

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.     

Locally G-homogeneous Busemann G-spaces

We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological n-manifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.     

Equivariant Brauer and Picard groups and a Chase–Harrison–Rosenberg exact sequence

The classical Chase–Harrison–Rosenberg exact sequence relates the Picard and Brauer groups of a Galois extension S of a commutative ring R to the group cohomology of the Galois group. We associate to each action of a locally compact group G on a locally compact space X two groups which we call the equivariant Picard group and the equivariant Brauer group. We then prove an analogue of the Chase–Harrison–Rosenberg exact sequence in the which the roles of the Picard and Brauer groups are played by their equivariant analogues.     

Chern character for totally disconnected groups

In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum–Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups.     

Shape Morphisms to Transitive G-Spaces

The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism F:X Y of a topological space X to a topological space Y is generated by a continuous mapping f:X Y. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive G-spaces, where G is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a G-space X to be equal to the group G itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.     

G-compactifications of pseudocompact G-spaces

Examples of a pseudocompact (even countably compact) G-space which is not G-Tychonoff and of a locally compact pseudocompact (even countably compact) G-Tychonoff space X with βGX≠βX are constructed.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

The wΔ-space problem

We address the following question: “Must every wΔ-space with a G_δ-diagonal be developable?” Consistently, the answer is “no.”Example. Assume CH. There is a zero-dimensional, scattered, locally compact, wΔ-space with a G_δ-diagonal which is not developable.For normal, locally compact spaces (or slightly weaker), the answer is “yes”.Theorem. If X is ω-sCWH, locally Lindelöf, wΔ-space with a G_δ-diagonal, then X is developable.     

On the Zeros of Krasner Analytic Functions

Given a prime number p and a compact subset X of the topological completion ? _p of the algebraic closure of the field of p-adic numbers, we study the set of zeros of Krasner analytic functions defined on ?¹(? _p ) \ X with values in ? _p .