Amenable Groups Research Articles

Local Hilbert–Schmidt stability

We introduce a notion of local Hilbert–Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert–Schmidt stable but not Hilbert–Schmidt stable. For amenable groups, we provide a criterion for local Hilbert–Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert–Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert–Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert–Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.

Medicinally Significant Enantiopure Compounds from Garcinia Acid Isolated from Garcinia gummi-gutta

Garcinia gummi-gutta, commonly known as Garcinia cambogia (syn.), is a popular traditional herbal medicine known for its role in treating obesity, and has been incorporated into several nutraceuticals globally for this purpose. The fruit rind is also used as a food preservative and a condiment because of its high content of hydroxycitric acid, which imparts a sharp, sour flavour. This review highlights the major bioactive compounds present in the tree Garcinia gummi-gutta, with particular emphasis on (2S, 3S)-tetrahydro-3-hydroxy-5-oxo-2,3-furan dicarboxylic acid, commonly referred to as garcinia acid. This acid can be isolated in large amounts through a simple procedure. Additionally, it explores the synthetic transformations of garcinia acid into biologically potent and functionally useful enantiopure compounds, a relatively under-documented area in the literature. This acid, with its six-carbon skeleton, a γ-butyrolactone moiety, and two chiral centres bearing chemically amenable functional groups, offers a versatile framework as a chiron for the construction of diverse molecules of both natural and synthetic origin. The synthesis of chiral 3-substituted and 3,4-disubstituted pyrrolidine-2,5-diones, analogues of the Quararibea metabolite—a chiral enolic-γ-lactone; the concave bislactone skeletons of fungal metabolites (+)-avenaciolide and (−)-canadensolide; the structural skeletons of the furo[2,3-b]furanol part of the anti-HIV drug Darunavir; (−)-tetrahydropyrrolo[2,1-a]isoquinolinones, an analogue of (−)-crispine A; (−)-hexahydroindolizino[8,7-b]indolones, an analogue of the naturally occurring (−)-harmicine; and furo[2,3-b]pyrroles are presented here.

On chaotic dynamics of the minimal centre of attraction of discrete amenable group actions

Let X be a compact metric space, G be a countable discrete infinite amenable group continuously acting on X and F be a Følner sequence of G. In this paper, we acquire some ergodic properties of the discrete amenable group action ( X , G ) . By virtue of these properties, we prove the existence of syndetic sensitivity of the minimal F -centre of attraction of a point x ∈ X provided that the minimal F -centre of attraction of x is not S-generic and admits a dense set of quasi-weakly almost periodic points relative to F of ( X , G ) . The presented results enhance some main results of [Z. Chen and X. Dai, Chaotic dynamics of minimal centre of attraction of discrete amenable group actions, J. Math. Anal. Appl. 456 (2017), pp. 1397–1414.].

Torsion homology growth and cheap rebuilding of inner-amenable groups

We prove that virtually torsion-free, residually finite groups that are inner-amenable and non-amenable have the cheap 1-rebuilding property, a notion recently introduced by Abért, Bergeron, Frączyk and Gaboriau. As a consequence, the first \ell^{2} -Betti number with arbitrary field coefficients and log-torsion in degree 1 vanish for these groups. This extends results previously known for amenable groups to inner-amenable groups. We use a structure theorem of Tucker-Drob for inner-amenable groups showing the existence of a chain of q -normal subgroups.

Van der Corput’s difference theorem for amenable groups and the left regular representation

We establish a connection between two variants of van der Corput’s Difference Theorem (vdCDT) for countably infinite amenable groups G and the ergodic hierarchy of mixing properties of a unitary representation U of G. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Minimal amenable subshift with full mean topological dimension

Abstract Let G be an infinite countable amenable group and P a polyhedron with the topological dimension dim ⁡ ( P ) < ∞ . We construct a minimal subshift (X, G) of ( P G , G ) such that its mean topological dimension is equal to dim ⁡ ( P ) . This result answers the question of Dou (2017 Discrete Contin. Dyn. Syst. 37 1411–24). Moreover, it extends the work of Jin and Qiao (2023 arXiv:2102.10339) for Z -action.

Universal inequalities for Dirichlet eigenvalues on discrete groups

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary condition on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Pólya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d -regular tree (simple random walk on the free group).

Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K \mathbb {K} . To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K \mathbb {K} -amenability, of which we prove an algebraic characterization. It implies that normed K \mathbb {K} -amenable groups are locally elliptic, and it relates an invariant, the norm of a K \mathbb {K} -amenable group, to the order of its discrete finite p p -subquotients, where p p is the characteristic of the residue field of K \mathbb {K} . Moreover, we prove a characterization of discrete normed K \mathbb {K} -amenable groups in terms of vanishing of bounded cohomology with coefficients in K \mathbb {K} . The algebraic characterization shows that normed K \mathbb {K} -amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K \mathbb {K} coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K \mathbb {K} has positive characteristic or its residue field has characteristic 0 0 , then the comparison map is injective in all degrees. If K \mathbb {K} is a finite extension of Q p \mathbb {Q}_p , we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2 2 . A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg–Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer–Vietoris sequence, the main missing piece for computing real bounded cohomology.

The pressure of intricacy and average sample complexity for amenable group actions

The pressure of intricacy and average sample complexity for amenable group actions

On sofic approximations of non amenable groups

In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a question of Pǎunescu and generalizes the Elek–Szabo uniqueness theorem for sofic approximations.

On the Garden of Eden theorem for non-uniform cellular automata

We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.

An example of an infinite amenable group with the ISR property

An example of an infinite amenable group with the ISR property

Simplicial volume of manifolds with amenable fundamental group at infinity

Abstract We show that for $n \neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

Variational principle of higher dimension weighted pressure for amenable group actions

Let r≥2 and (Xi,G)(i=1,⋯,r) be topological dynamical systems with an infinite countable discrete amenable phase group G. Suppose that πi:(Xi,G)→(Xi+1,G) are factor maps, a={a1,⋯,ar−1}∈Rr−1 is a vector with 0≤ai≤1 and (w1,⋯,wr)∈Rr is a probability vector associated with a. In this paper, given f∈C(X1), we introduce the weighted topological pressure Pa(f,G). Moreover, by using measure-theoretical theory, we establish a variational principle:Pa(f,G)=supμ∈MG(X1)⁡(∑i=1rwihμi(Xi,G)+w1∫X1fdμ), where h{⋅}(⋅,G) is the Kolmogorov-Sinai entropy of the systems acted by the amenable group G and μi=πi−1∘⋯∘π1μ is the induced G-invariant measure on Xi.

Non-existence of a universal zero-entropy system via generic actions of almost complete growth

Abstract We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

New non-local mean methods for MRI denoising based on global self-similarity between values

Magnetic resonance imaging (MRI) is a non-invasive medical imaging technique that provides high-resolution 3D images and valuable insights into human tissue conditions. Even at present, the refinement of denoising methods for MRI remains a crucial concern for improving the quality of the images. This study aims to improve the prefiltered rotationally invariant non-local principal component analysis (PRI-NL-PCA) algorithm. We relaxed the original restrictions using particle swarm optimization to determine optimal parameters for the PCA part of the original algorithm. In addition, we adjusted the prefiltered rotationally invariant non-local mean (PRI-NLM) part by traversing the signal intensities of voxels instead of their spatial positions to reduce duplicate calculations and expand the search volume to the whole image when estimating voxels’ signal intensities. The new method demonstrated superior denoising performance compared to the original approach. Moreover, in most cases, the new algorithm ran faster. Furthermore, our proposed method can also be applied to process Gaussian noise in natural images and has the potential to enhance other NLM-based denoising algorithms.

Some remarks on summability defined by invariant mean in intuitionistic fuzzy 2-normed spaces

In present paper, we aim to define a new summability method using σ-mean called σ-statistical summability in an intuitionistic fuzzy 2- normed space (briefly IF2NS). We also define σ-statistical cauchy sequence in an IF2NS and study some of their properties. We display example that shows our method of summability is more stronger in these spaces.

Entropy, virtual Abelianness and Shannon orbit equivalence

Abstract We prove that if two free probability-measure-preserving (p.m.p.) ${\mathbb Z}$ -actions are Shannon orbit equivalent, then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein–Weiss and the entropy invariance results of Austin and Kerr–Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every ${\mathbb Z}$ -odometer is Shannon orbit equivalent to the universal ${\mathbb Z}$ -odometer.