Yair Glasner, Nicolas Monod
We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property.
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Let $G$ be an amenable group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift then $X$ has the Myhill property, that is, every pre-injective cellular automaton $τ\colon X \to X$ is surjective.
Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $τ\colon A^S \to A^S$ is surjective.
Let $X$ be a compact metrizable space equipped with a continuous action of a countable amenable group $G$. Suppose that the dynamical system $(X,G)$ is expansive and is the quotient by a uniformly bounded-to-one factor map of a strongly irreducible subshift. Let $τ\colon X \to X$ be a continuous map commuting with the action of $G$. We prove that if there is no pair of distinct $G$-homoclinic points in $X$ having the same image under $τ$, then $τ$ is surjective.
A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.
We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.
We construct inner amenable groups G with infinite conjugacy classes and such that the associated II_1 factor does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.
The notions of a box family and fibred cofinitely-coarse embedding are introduced. The countable, residually amenable groups satisfying the Haagerup property are then characterized as those possessing a box family that admits a fibred cofinitely-coarse embedding into a Hilbert space. This is a generalization of a result of X. Chen, Q. Wang and X. Wang on residually finite groups.
Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more genera…
Amenable equivalence relations, Kesten's property, and measurable lamplighters
本文研究了可数Borel等价关系的可 amenability 性质，并通过凯斯滕性质与可测灯塔群的反集中不等式建立了联系，从而构造了一个无凯斯滕性质的可 amenable 可缩波兰群。
This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer.
We define a probability measure preserving and r-discrete groupoid that is associated to every invariant point process on a locally compact and second countable group. This groupoid governs certain factor processes of the point process, in particular the existence of Cayley factor graphs. With this method we are able to show that point processes on amenable groups admit all (and only admit) Cayley factor graphs of amenable groups, and that the Poisson point process on groups with Kazhdan's Property (T) admits no Cayley factor graphs. This gives examples of pmp countable Borel equivalence relat…
We show that, if $G$ is an amenable group and $H$ is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$.
A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal
本文构造了Caprace非内可均群的有限生成扩张，该扩张为GL_4(F_2(t))的ICC子群，无性质(T)，且非适当近端，从而解决了Boutonnet等人提出的问题。
We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology.
We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…
RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions
本文研究了可算群作用下C*-代数的交叉积及可算群体C*-代数的RFD性质，通过动作特征化RFD属性，并推广Bekka定理至动力学方向。
In this paper we give sufficient conditions for a free product of residually finite groups to admit an embeddable box space. This generalizes the constructions of Arzhantseva, Guentner and Spakula in arXiv:1101.1993v2 and gives a new class of non-amenable metric spaces with bounded geometry which coarsely embeds into Hilbert space.
For an amenable minimal topologically free dynamical system $α$ of a group on a compact metrizable space $Z$ and for a compact metrizable space $Y$ satisfying a mild condition, we construct a minimal skew product extension of $α$ on $Z\times Y$. This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.
Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property O. It is deduced both that these groups are exact and satisfy Yu's Property A.
We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over $\mathbb{S}^1$.
We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. García-Ramos and H. Li, and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is pre-injective. This together with an earlier result of…
For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T) admits a faithful …
We give a simple and unified proof showing that the unrestricted wreath product of a weakly sofic, sofic, linear sofic, or hyperlinear group by an amenable group is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. By means of the Kaloujnine-Krasner theorem, this implies that group extensions with amenable quotients preserve the four aforementioned metric approximation properties. We also discuss the case of co-amenable groups.
We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that families of independent uniformly distributed permutation matrices and certain families of non-random permutation matrices (essentially, those coming from quasi--actions of a sofic group) are asymptotically *-free as the matrix size grows without bound.
Amenable wreath products with non almost finite actions of mean dimension zero
本文通过研究顺从圈积群的变构性，构造出平均维数为零但不具几乎有限性的顺从群极小拓扑自由作用，证明Kerr的Toms-Winter猜想动力学对应结论对这类作用不成立，并给出首批顺从变构群例子。
We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.
We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of amenable groups and the amalgamated free products of two amenable groups over a finite subgroup admit such actions.
A character approach to the ISR property
本研究提出研究ISR性质的特征标方法，引入不可分解正则特征标性质推导相关群的ISR性质，证明近似有限群具有ISR性质，并构造出两类新的具有ISR性质的非顺从群。
Continuum of allosteric actions for non-amenable surface groups
本文针对非顺从闭曲面基本群，构造了连续统个具有拓扑自由但非本质自由性质的 p.m.p. 遍历极小 profinite 作用，并证明所得 IRS 两两不同。
Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence.
We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, …
Liouville property of actions of discrete groups can be reformulated in terms of existence co-F$ø$lner sets. Since every action of amenable group is Liouville, the property can be served as an approach for proving non-amenability. The verification of this property is conceptually different than finding a non-amenable action. There are many groups that are defined by strongly transitive actions. In some cases amenability of such groups is an open problem. We define $n$-Liouville property of action to be Liouville property of point-wise action of the group on the sets of cardinality $n$. We refo…
Self-simulability of graph products
研究图积的自模拟性，证明无限有限生成群的图积自模拟当且仅当其定义图没有由可解群组成的分隔团，并应用于图积在可解子群上的分裂。
We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.
We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.
Liouville property for groups and conformal dimension
本文证明：若迭代单值群的底层空间共形维数小于2，则该群上所有有限二阶矩对称随机游走具有Liouville性质，从而群可均，并应用于次双曲有理函数的Julia集。
Property FW is a natural combinatorial weakening of Kazhdan's Property T. We prove that the group of piecewise homographic self-transformations of the real projective line, has "few" infinite subgroups with Property FW. In particular, no such subgroup is amenable or has Kazhdan's Property T. These results are extracted from a longer paper. We provide a complete proof, whose main tools are the use of the notion of partial action and of one-dimensional geometric structures.
We prove that the class of residually C groups is closed under taking graph products, provided that C is closed under taking subgroups, finite direct products and that free-by-C groups are residually C. As a consequence, we show that local embeddability into various classes of groups is stable under graph products. In particular, we prove that graph products of residually amenable groups are residually amenable, and that locally embeddable into amenable groups are closed under taking graph products.
We prove non-amenability of the product replacement graphs Γ_n(G) for uniformly non-amenable groups. We also prove it for Z-large groups, when n is sufficiently large. It follows that Γ_n(G) is non-amenable when n is sufficiently large for hyperbolic groups, linear groups, and elementary amenable groups.
Hyperlinearity via amenable near representations
本文通过研究可 amenable近表示，证明了群超线性性的充要条件，并回答了Pestov和Kwiatkowska的问题。
We introduce the weak Haagerup property for locally compact groups and prove several hereditary results for the class of groups with this approximation property. The class contains a priori all weakly amenable groups and groups with the (usual) Haagerup property, but examples are given of groups with the weak Haagerup property which are not weakly amenable and do not have the Haagerup property. In the second part of the paper we introduce the weak Haagerup property for finite von Neumann algebras, and we prove several hereditary results here as well. Also, a discrete group has the weak Haageru…
Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-index subgroup of Gamma has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell.
Examples of non-amenable, boundary-amenable dynamical systems
研究具有(AP)性质的离散可数群$\Gamma$的作用，当它以特定方式作用于可数集时，在$\partial_\beta \mathfrak{X}$上诱导作用拓扑可和，给出了在$\partial_\beta (\Gamma / \Lambda)$上作用的多种情形及相关应用。
The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In this Part I, we prove that a disjoint union has property A of G. Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems…
It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, ν)$ factors onto $G \curvearrowright (X, μ)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G \curvearrowright (X, μ)$ has infinite entropy, then the action $G \curvearrowright (Y, ν)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \curvearro…
We study lattice embeddings for the class of countable groups $Γ$ defined by the property that the largest amenable uniformly recurrent subgroup $A_Γ$ is continuous. When $A_Γ$ comes from an extremely proximal action and the envelope of $A_Γ$ is co-amenable in $Γ$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $Γ$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.
On operator Connes-amenability of the Fourier-Stieltjes algebra
本文研究了傅里叶-斯蒂尔杰斯代数B(G)的算子康尼斯-可 amenability性质，通过构造反例推翻了关于B(G)与群G可 amenability关系的猜想。
We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in $(\bZ,+)$, valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study.
We define a strengthening of the Haagerup-Kraus approximation property by means of the subalgebras of Herz-Schur multipliers $M_d(G)$ ($d\geq 2$) introduced by Pisier. We show that unitarisable groups satisfying this property for all $d\geq 2$ are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy $M_d$-AP for all $d\geq 2$. We also give examples of non-weakly amenable groups satisfying $M_d$-AP for all $d\geq 2$.
McDuff factors from amenable actions and dynamical alternating groups
本文证明可均群在康托集上的拓扑自由作用产生的包含交错群的子群，其群冯诺依曼代数为McDuff因子，并推广到广义圈积的稳定性。
We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-sta…
A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group.
A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, inc…
Cohomological and quasi-isometric diversity of groups with property $R_\infty$
本文研究具有性质$R_\infty$的群的上同调与准等距多样性，证明了此类群在强意义上非常丰富，存在不可数多个有限生成群具有$R_\infty$且彼此不准等距。
We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in part…
We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class $\mathcal{S}$ groups. As consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups $A \wr G$, $B \wr Γ$ of non-amenable exact direct product groups $G$, $Γ$ with amenable bases $A$, $B$ are measure equivalent, then $G$ and $Γ$ are measure equivalent. We get Bass--Serr…
We calculate the rank gradient and p-gradient of free products, free products with amalgamation over an amenable subgroup, and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For the p-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used.
Weak amenability of discrete groups was introduced by Haagerup and co-authors in the 1980's. It is an approximation property known to be stable under direct products and free products. In this paper we show that graph products of weakly amenable discrete groups are weakly amenable (with Cowling-Haagerup constant 1). Along the way we construct a wall space associated to the word length structure of a graph product and also give a method for extending completely bounded functions on discrete groups to a completely bounded function on their graph product.
M. Beiglböck, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of $% EAB$. When $G=\…
The algebraic cheap rebuilding property
本文提出公理化框架处理群和链复形的同调性质组合定理，引入代数廉价重建性质，证明某些图群挠同调增长消失。
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