Amenable Groups Research Articles

On the bounded cohomology for ergodic nonsingular actions of amenable groups

Let Γ be an amenable countable discrete group. Fix an ergodic free non-singular action of Γ on a nonatomic standard probability space. Let G be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of G is compact in the natural topology. It is shown that there exists a bounded ergodic G-valued cocycle of Γ.

A hierarchy of topological systems with completely positive entropy

We define a hierarchy of systems with topological completely positive entropy in the context of countable amenable continuous group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.

An Alternative Definition of Topological Entropy for Amenable Group Actions

An Alternative Definition of Topological Entropy for Amenable Group Actions

Closed [formula omitted]-structures on nilmanifolds

In this paper we consider closed SL(3,C)-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to G2-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed SL(3,C)-structure and those which admit an invariant mean convex half-flat SU(3)-structure. We also prove that, if a solvmanifold admits an invariant tamed closed SL(3,C)-structure, then it also has an invariant symplectic half-flat SU(3)-structure.

Dynamical intricacy and average sample complexity of amenable group actions

Dynamical intricacy and average sample complexity of amenable group actions

Metric Versus Topological Receptive Entropy of Semigroup Actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

Amenable upper mean dimensions

In this paper we formulate the notions of upper metric mean dimension and upper measure-theoretic mean dimension for amenable group actions. In particular, a different version of variational principle for amenable group actions is presented by replacing the condition on the space in Velozo and Velozo (Rate distortion theory, metric mean dimension and measure theoretic entropy. Preprint ArXiV: 1707.05762 [Math.DS], 2017) by the conditions on the systems. At last, we discuss the relation between upper metric mean dimensions and pseudo-orbits.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

AbstractWe prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

The Proximal Relation, Regionally Proximal Relation and Banach Proximal Relation for Amenable Group Actions

In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions. A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation, and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Pr\"ufer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

Complexity among the finitely generated subgroups of Thompson's group

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group F which is strictly well-ordered by the embeddability relation of type \epsilon_0 +1 . All except the maximum element of this family (which is F itself) are elementary amenable groups. In fact we also obtain, for each \alpha < \epsilon_0 , a finitely generated elementary amenable subgroup of F whose EA-class is \alpha + 2 . These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with \mathbf Z + \mathbf Z , \mathbf Z \wr \mathbf Z , and the Brin–Navas group B . We also give an example of a pair of finitely generated elementary amenable subgroups of F with the property that neither is embeddable into the other.

CAT(0) cube complexes and inner amenability

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group G on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty G -invariant closed convex subset such that every conjugation invariant mean on G gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on G . We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to “locate” conjugation invariant means on a group G relative to a given normal subgroup N of G . We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

Residual finiteness for central pushouts

We prove that pushouts $A*_CB$ of residually finite-dimensional (RFD) $C^*$-algebras over central subalgebras are always residually finite-dimensional provided the fibers $A_p$ and $B_p$, $p\in \mathrm{spec}~C$ are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group $C^*$-algebras. Along the way, we discuss the problem of when, given a central group embedding $H\le G$, the resulting $C^*$-algebra morphism is a continuous field: this is always the case for amenable $G$ but not in general.

Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature

For Riemannian submersions with fibers of basic mean curvature, we compare the spectrum of the total space with the spectrum of a Schrödinger operator on the base manifold. Exploiting this concept, we study submersions arising from actions of Lie groups. In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups.

Minimal systems with finitely many ergodic measures

In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal equicontinuous factor. This result is obtained as an application of a general criterion which states that if a minimal system under an amenable group action is an almost finite to one extension of its maximal equicontinuous factor and has no infinite independent sets of length k for some k≥2, then it has only finitely many ergodic measures.

Groups with infinite FC-center have the Schmidt property

Abstract We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

Ubiquity of entropies of intermediate factors

We consider topological dynamical systems $(X,T)$, where $X$ is a compact metrizable space and $T$ denotes an action of a countable amenable group $G$ on $X$ by homeomorphisms. For two such systems $(X,T)$ and $(Y,S)$ and a factor map $\pi : X \rightarrow Y$, an intermediate factor is a topological dynamical system $(Z,R)$ for which $\pi$ can be written as a composition of factor maps $\psi : X \rightarrow Z$ and $\varphi : Z \rightarrow Y$. In this paper we show that for any countable amenable group $G$, for any $G$-subshifts $(X,T)$ and $(Y,S)$, and for any factor map $ \pi :X \rightarrow Y$, the set of entropies of intermediate subshift factors is dense in the interval $[h(Y,S), h(X,T)]$. As a corollary, we also prove that if $(X,T)$ and $(Y,S)$ are zero-dimensional $G$-systems, then the set of entropies of intermediate zero-dimensional factors is equal to the interval $[h(Y,S), h(X,T)]$. Our proofs rely on a generalized Marker Lemma that may be of independent interest.

Weakly almost periodic and uniformly continuous functionals on the Orlicz Figà-Talamanca Herz algebras

In this paper we study weakly almost periodic and uniformly continuous functionals on the Orlicz Figà-Talamanca Herz algebras associated to a locally compact group. We show that a unique invariant mean exists on the space of weakly almost periodic functionals. We also characterise discrete groups in terms of the inclusion of the space of uniformly continuous functionals inside the space of weakly almost periodic functionals.

Predictability, topological entropy, and invariant random orders

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. We investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.

Convergence of almost orbits of semigroups

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of $${\ell }^\infty (S)$$ which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let $$(E, \Vert \cdot \Vert _E)$$ be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and $$\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}$$ is an asymptotically nonexpansive semigroup which acts on C. Assume also that $$\mathcal {T}$$ has a unique fixed point $$\tilde{x}$$ and that, in addition, this point $$\tilde{x}$$ lies on the boundary $$\partial C$$ of C. Then $$\{u(s) \}_{s \in S}$$ converges strongly to $$\tilde{x}$$ for each almost orbit $$u \in AO_Y (\mathcal {T})$$ .