Conjugacy class conditions in locally compact second countable groups

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 present a theory of continuous bounded cohomology of locally compact groups with coefficients in Banach modules. A central role is played by amenable actions, as they give rise to relatively injective resolutions. Further, we propose a substitute for the Mautner property, based on the virtual subgroup viewpoint, and we show (Theorem 6) that all compactly generated locally compact groups, e.g. finitely generated groups, satisfy it. This, together with the cohomological characterization of amenable actions, leads to a refined version of a higher degree Lyndon-Hochschild-Serre exact sequence (Theorem 13), which entails a stronger Kunneth type formula for continuous bounded cohomology in degree two. We apply this theory to general irreducible lattices in products of lo- cally compact groups: we obtain notably super-rigidity results for bounded cocycles (Theorem 16 and Corollary 23), rigidity results for actions by diffeomorphisms on the circle (Corollary 22) and vanishing of the stable commutator length (Corollary 32). More applications will be published elsewhere. In the spirit of relative homological algebra, we give for a locally com- pact second countable group G a functorial characterization of the contin- uous bounded cohomology of G with coefficients. The resolutions and the notion of relatively injective objects (Defini- tion 1.4.2) are set up in the category of continuous Banach G-modules, while the coefficients are mainly duals of separable continuous Banach G- modules (henceforth called coefficient modules), including notably separa- ble continuous unitary representations, L ∞ spaces and trivial coefficients. We emphasize that on all Banach G-modules, the G-action is isometric. If E is a coefficient module and S a regular measure G-space (see Defi- nition 1.3.1), let L ∞∗ (S, E) be the space of weak-* measurable essentially bounded maps; we consider the resolution 0 E d L ∞∗(S, E) d L ∞∗ (S 2 ,E ) d L ∞∗ (S 3 ,E ) d ···

A countable group G has the strong topological Rokhlin property (STRP) if it admits a continuous action on the Cantor space with a comeager conjugacy class. We show that having STRP is a symbolic dynamical property. We prove that a countable group G has STRP if and only if certain sofic subshifts over G are dense in the space of subshifts. A sufficient condition is that isolated shifts over G are dense in the space of all subshifts.We provide numerous applications including the proof that a group that decomposes as a free product of finite or cyclic groups has STRP. We show that finitely generated nilpotent groups do not have STRP unless they are virtually cyclic; the same is true for many groups of the form G_{1}\times G_{2}\allowbreak\times\nobreak G_{3} where each factor is recursively presented. We show that a large class of non-finitely-generated groups do not have STRP; this includes any group with infinitely generated center and the Hall universal locally finite group.We find a very strong connection between STRP and shadowing, also called the pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group G if and only if G has STRP.

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 $

Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex, involutive structures on compact Lie groups can be computed by using only Lie algebras, thus reducing the analytical problem to a purely algebraic one. The main tool is the Leray spectral sequence that connects the result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault cohomology of a homogeneous manifold.

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.

We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group A A and a group B B of topological group automorphisms of A , A A, A has small B B invariant neighborhoods at 1 if and only if A ^ \hat A has precompact orbits under the dual actions of B B .

Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.

In this paper we consider a locally compact second countable unimodular group G and a closed unimodular subgroup H. Let ρ be a finite-dimensional unitary representation of H with closed image. For the unitary representation of G obtained by inducing ρ from H to G a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called (ρ,H) spherical distributions, which leads to a description of the decomposition on the level of these distributions