Actions of finitely generated groups on the universal Menger curve

We prove that every finitely generated group acts effectively on the universal Menger curve.

     

On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .

     

Topological equivalence of Haar measures on compact groups

The existence of homeomorphisms establishign an isometry of normalized Haar measures on (metrizable) compact groups is studied. In the case of 0-dimensional groups, a complete answer is given in terms of the indices of open normal subgroups. For example, for the countable powers of the groups ℤ/(m) and ℤ/(n), the answer is affirmative if and only ifm andn have the same prime divisors. A certain class of extensions of 0-dimensional groups is also studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 188–194, August, 2000.     

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.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Compact quantum metric spaces and ergodic actions of compact quantum groups

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T→EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of γ in each fibre is continuous over T for every equivalence class γ of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podle? spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology.     

Expansiveness of algebraic actions on connected groups

We study endomorphism actions of a discrete semigroup on a connected group . We give a necessary and sufficient condition for expansiveness of such actions provided is either a Lie group or a solenoid.

     

Nonlinearizable actions of dihedral groups on affine space

Let be a reductive, non-abelian, algebraic group defined over . We investigate algebraic -actions on the total spaces of non-trivial algebraic -vector bundles over -modules with great interest in the case that is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.

     

G-spaces and Fixed Point Sets

For a compact connected group G, in a compact free G-space M every closed subset is the fixed point set of an autohomeomorphism if the weight of M is not bigger than that of G. In a compact group with nonsingleton components, every closed subset is the fixed point set of an autohomeomorphism if and only if the weight of the whole group is not bigger than that of the component of the identity.     

Fixed point property and normal structure for Banach spaces associated to locally compact groups

In this paper we investigate when various Banach spaces associated to a locally compact group have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a -algebra has the fixed point property then for any normal element of , the spectrum is countable and that the group -algebra has weak normal structure if and only if is finite.

     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

Robustly transitive actions of ℝ<Superscript>2</Superscript> on compact three manifolds

In this paper, we define robust transitivity for actions of ℝ² on closed connected orientable manifolds. We prove that if the ambient manifold is three dimensional and the dense orbit of a robustly transitive action is not planar, then the action is defined by an Anosov flow, i.e. its orbits coincide with the orbits of an Anosov flow.     

Continuous spectral decompositions of Abelian group actions on -algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual group of G as continuous spectral decompositions of G-actions on C^*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C^*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.     

Orthogonal wavelets on direct products of cyclic groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p ⁿ numerical coefficients generate multiresolution analyses in L ²(G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p ⁿ parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L ²(G) is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.     

Automatic surjectivity of ring homomorphisms on -algebras and algebraic differences among some group algebras of compact groups

In this paper we present two automatic surjectivity results concerning ring homomorphisms between -classes of an -algebra which, in some sense, improve the main theorem in a recent paper by the author (Proc. Amer. Math. Soc. 124 (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups , no quotient ring of is isomorphic to , a statement we conjecture to be true for every pair of group rings corresponding to different exponents .

     

On Finite Groups Acting on Contractible Complexes of Dimension Two

We show that every action of a finite group on a one-connected diagrammatically reducible two-complex has a fixed point. The proof makes use of Stallings' notion of foldings of graphs, appropriately adapted to a certain two-dimensional setting.     

Chern and Pontryagin Numbers in Perfect Symmetries of Spheres

The paper presents a procedure for constructing smooth actions of finite perfect groups on spheres with fixed point sets having certain prescribed properties (Theorem A); in particular, having any prescribed configuration of Chern and Pontryagin numbers (Corollary C). The main ingredients used are equivariant thickening and equivariant surgery.