Continuous bounded cohomology and applications to rigidity theory

Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis double-struck upper Q comma greater-than right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Aut(\mathbb {Q},&gt;)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and, less trivially, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis script upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Aut(\mathcal {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the random graph. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.

The second degree cohomology of finite orthogonal groups, II

We consider a countable discrete group Γ \Gamma acting ergodically on a standard Borel space S with quasi-invariant measure μ \mu . Let π \pi be a unitary representation of Γ \Gamma on L 2 ( S , d μ , H ) {L^2}(S,d\mu ,\mathcal {H}) "nicely" related with S. We prove that if Γ \Gamma acts amenably on S then π \pi is weakly contained in the regular representation.

We prove that the Poisson boundary of any spread out non-degenerate symmetric randomwalk on an arbitrary locally compact second countable group G is doubly $\mathcal{M}$sep-ergodic with respect to the class $\mathcal{M}$sep of separable coefficient Banach G-modules. The proof is direct and based on an analogous property of the bilateral Bernoulli shift in the space of increments of the random walk. As a corollary we obtain that any locally compact s-compact group G admits a measure class preserving action which is both amenable and doubly $\mathcal{M}$sep-ergodic. This generalizes an earlier result of Burger and Monod obtained under the assumption that G is compactly generated and allows one to dispose of this assumption in numerous applications to the theory of bounded cohomology.

Fixed points for bounded orbits in Hilbert spaces

We provide a proof that the vanishing of $\ell^2$-Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence.

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ <i>disperses</i> if $\lim_

Totally Lindelöf and totally ω-narrow paratopological groups

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $\sigma $-finite standard measure space $

We study homoclinic points of non-expansive automorphisms of compact abelian groups. Connections between the existence of non-trivial homoclinic points, expansiveness, entropy and adjoint automorphisms (in the sense of Einsiedler and Schmidt) are explored. Some implications for countable abelian group actions by automorphisms of compact abelian groups are also considered and it is shown that if every element of such an action has finite entropy, there can be no non-trivial common homoclinic points, unless the action is generated by a single automorphism.

Introduction Let f be a homomorphism from a second countable locally compact group G into the continuous automorphisms of a second countable compact group K. Let g ∗ k = f(g)(k) and suppose further the mapping given by (g, k) → g ∗ k is continuous from G × K into K, g ∈ G, k ∈ K. Then the cartesian product K × G becomes a group with multiplication given by (k1, g1)(k2, g2) = (k1(g1 ∗ k2), g1g2), where (ki, gi) ∈ K × G, i = 1, 2. We denote this semi-direct product group by K ×s G. Let (S, μ), a standard Borel space, be an ergodic Borel K ×s G-space with a probability invariant measure μ. In this paper we state necessary and sufficient conditions so the quotient Borel K ×s G-mapping from S into the space of K-orbits in S has relative discrete spectrum. These conditions are in terms of the stabilizers and the natural action of G on the dual K of K. (We remind the reader K is the set of unitary equivalence classes of irreducible unitary representations of K). Choose a Borel subset S of S that meets each K-orbit in S exactly once. Let p : (S, μ) → (S, μ) be a mapping given by p(s) = s where s is the unique point in S that meets the K-orbit s · K of s and μ = μ ◦ p−1.

In a preceding article~[7] the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

A 5-term exact sequence and the interpretation of low dimensional groups of homology with trivial coefficients of algebras with bracket are obtained. The endofunctor in the category of algebras with bracket which assigns to a perfect algebra with bracket its universal central extension is constructed and the characterization of the universal central extension by means of the low-dimensional homology groups is done. The conditions required to lift an automorphism or a derivation of A to A′ in a covering f:A′ ↠ A are analyzed.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.