Azer Akhmedov
We present new metric criteria for non-amenability and discuss applications. The main application of the results of this paper is the proof of non-amenability of R.Thompson's group F. This is a continuation of the series of papers on our criteria for non-amenability. The current paper is independent of other papers in the series.
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Amenability problem for Thompson's group $F$: state of the art
本文综述了Thompson群$F$顺从性问题的近期研究成果，主要涉及Cayley图有限子图密度估计及群环方程，并探讨了解决该问题的双向途径。
We present a proof of non-amenability of R.Thompson's group F.
The goal is to extend a non-standard amenability test for groups, based on random walks and superharmonic functions, to group actions on sets, and to apply it to Thompson's group F using certain properties of extensive amenability. While no conclusive answer regarding the amenability of F is given, the approach is helpful in developing a new potentially useful criterion and testing it on a significant subclass of superharmonic functions.
In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $Γ$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.
Non-inner amenability of the Higman-Thompson groups
本文证明了 Higman-Thompson 群 $T_n$ 和 $V_n$ 对所有 $n\ge 2$ 均非内 amenable，推广了 Haagerup-Olesen 在 $n=2$ 时的结果，并采用不同于 Thurston 分片射影模型的新工具，同时为原结果提供了替代证明。
This paper examines Shavgulidze's postings to the ArXiv and publications which concern his argument that Thompson's group F is amenable. In particular I list specific places where there are errors in each of the postings and publications and give a public clarification of some of the speculation around these papers.
In this paper we prove that the general version, F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P.Jolissaint. To do so, we prove first that F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property $Γ$ for the group von Neumann algebras involved. By a result of E.G.Effros, F(N) follows inner amenable.
Recently Akhmedov introduced a property of finitely generated groups called the Traveller Salesman Property (TSP). It was shown by Thurston that TSP implies non-amenability. Akhmedov conjectured that R. Thompson's group $F$ has TSP. The aim of this article is to disprove this conjecture.
By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent.
The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.
In this paper we study the non-amenability question of the Thompson group $F$ from the $C^{*}$ algebra side. Using a characterization of amenability in this framework we set about evaluating the reduced norm of the averages $\frac{1}{n}\sum x_0^ix_1x_0^{-i}$, where $x_0$ and $x_1$ are the generators of $F$ in its finite presentation. We prove that when $n$ is sufficiently large the above norm concentrates on a specific subset of $F$, easy to describe using the new normal form for elements in $F$, found by Guba and Sapir. We view this subset as the only obstruction against non-amenability.
Notes that elaborate on the details of a Theorem of Shavgulidze that implies the amenability of R. J. Thompson's group F.
It is a long standing open problem whether the Thompson group $F$ is an amenable group. In this paper we show that if $A$, $B$, $C$ denote the standard generators of Thompson group $T$ and $D:=C B A^{-1}$ then $$\sqrt2+\sqrt3\,<\,\frac1{\sqrt{12}}||(I+C+C^2)(I+D+D^2+D^3)||\,\le\, 2+\sqrt2.$$ Moreover, the upper bound is attained if the Thompson group $F$ is amenable. Here, the norm of an element in the group ring $\mathbb{C} T$ is computed in $B(\ell^2(T))$ via the regular representation of $T$. Using the "cyclic reduced" numbers $τ(((C+C^2)(D+D^2+D^3))^n)$, $n\in\mathbb{N}$, and some methods …
Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given.
Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given.
Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2. A discrete group G is amenable if there exists a finitely-additive probability measure on G which is invariant under left translations and is defined on all subsets of G. The amenability question for F is a well known open problem. In this paper we prove that group F is not amenable.
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph…
This paper allows one to obtain a criterion for the existence of a projectively invariant measure formulated in terms of combinatorial properties of a group (amenability of some canonical quotient group). Such necessary and sufficient condition is a basis of the classification scheme for groups of homeomorphisms of the line. In particular, a nonamenability of Thompson's group $F$ follows from the obtained criterion.
We find new presentations for the Thompson's groups $F$, the derived group $F^{'}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{'}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually finite dimensional.
A new proof of a Theorem of Nagnibeda and Pak on critical probabilities is given. The distinctive feature is to use a criterion of Folner which has been largely (and inexplicably) forgotten.
Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. The Ore condition says that for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It always holds whenever $G$ is amenable. Recently it was shown that for R.\,Thompson's group $F$ the converse is also true. So the famous amenability problem for $F$ is equivalent to the question on the Ore condition for the group ring of the same group. It is easy to see that the problem on the Ore condition for $K[F]$ is equivalent to the same property for the monoid ring $K[M]$, where $M$ is the monoid of positive e…
Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$ and to $$(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both cases the norm of an element in the group ring $\mathbb{C} F$ is computed in $B(\ell^2(F))$ via the regular representation of $F$. By extensive numerical computations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$, as well as good estimates of the spectral distributions of $(I+A…
We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of thi…
We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.
Frame Vector Group Representations and Amenability Properties
本文提出了一种基于框架表示的可度量性特征，定义了framenable群类，并展示了其与性质（T）的对比以及包含多种已知群的例子。
Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.
We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for…
This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation \[Δ= < b,t|[b,t^2]b^{-1},[[[b,t^{-1}],b],b]>.\] I show that it acts transitively on a 3-regular tree, and that $Γ=< b,b^{t^{-1}}$ stabilizes a vertex and acts by restriction on a binary rooted tree. $Γ$ is a "fractal group", generated by a 3-state automaton, and is a discrete analogue of the monodromy action of iterates of f(z)=z^2-1 on associated coverings of the…
The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are obtained for both conjectures. The paper will also contain some general analysis of the conjectures.
Numerical studies of Thompson's group F and related groups
本文研究了Thompson组及相关组的共增长级数，开发了多项式时间算法生成这些级数的项，并通过数值技术提取其渐进行为，给出了Thompson组F的共增长级数的改进下界，探讨了该组是否可 amenable 的问题。
Kesten's criterion for discrete probability measure-preserving groupoids
本文基于凯斯滕准则，通过研究对称不变马尔可夫算子的算子范数，提出离散概率测度保持群oid 可 amenability 的判据。
By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable iff the density of the corresponding Cayley graph equals $2m$. A famous problem on the amenability of R.\,Thompson's group $F$ is still open. What is known due to the result by Belk and Brown, is that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimat…
We observe that a Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as amenability at infinity of discrete countable groups. As corollary, the latter property can also be tested by actions on the Hilbert cube. These results generalize a criterion due to Giordano and de la Harpe.
Higher Brin--Thompson groups are Kazhdan
本文给出代数判据，证明高阶Brin--Thompson群nV在n较大时具有Kazhdan性质(T)，并统一证明nV、Aut(F_n)及EL_n(R)均具有性质(T)，附录还证明nV非超线性且非Schatten p-可近似。
Amenability of quadratic automaton groups
研究二次活动自动群的可 amenability，通过Schreier图的电导下界结合JNS结果证明其可 amenability，提出新的加权Nash-Williams准则。
We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). …
In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson's group $V$ is hyperbolically elementary, and we deduce that it satisfies Property $(FW_{\infty})$, ie., every isometric action of $V$ on a finite-dimensional CAT(0) cu…
The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenability constitutes a considerable weakening of the Folner criterion. As a by-product, it will be shown that in any non amenable group G, there is a subset E of G such that no finitely additive probability measure on G measures all translates of E equally.
Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. It is known that if $G$ is amenable then $R$ satisfies the Ore condition: for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It is also true for amenable groups that a non-zero solution exists for any finite system of linear equations over $R$, where the number of unknowns exceeds the number of equations. Recently Bartholdi proved the converse. As a consequence of this theorem, Kielak proved that R.\,Thompson's group $F$ is amenable if and only if it satisfies the Ore condition. The amenability pr…
We will give a general criterion - the existence of an $F$-obstruction - for showing that a subgroup of $\mathrm{PL}_+ I$ does not embed into Thompson's group $F$. An immediate consequence is that Cleary's "golden ratio" group $F_τ$ does not embed into $F$. Our results also yield a new proof that Stein's groups $F_{p,q}$ do not embed into $F$, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of $F$-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the …
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
Thompson-like groups, Reidemeister numbers, and fixed points
本文研究了类Thompson群$F$的自同构不动点性质，推导出一种上同调判据来检测阿贝尔化中的无限不动点集并蕴含性质$R_\infty$，利用BNS $Σ$-不变量证明该工具广泛适用于多种$F$类群。
We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $LΛ\subset LΓ$ arising from amenable subgroups $Λ$ of discrete countable groups $Γ$. The criterion is expressed in terms of $Λ$-invariant measures on some compact $Γ$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C*-algebra.
Recent work of Kaplan and Levy refining a nonsolvability criterion proved by Thompson in his N-Groups paper prompts questions on whether certain conditions on groups are equivalent to nonsolvability.
Abstract twisted Brin--Thompson groups
本文研究抽象扭曲布林-汤普森群的性质，证明其相对简单性，并展示其在有限生成简单群嵌入中的应用，同时探讨了相关猜想和新结果。
We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer …
In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for amenability. We observe that a Polish group is amenable if and only if its every continuous action on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as am…
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ an…
Discrete group extensions of countable Markov shifts and pressure gap
本文将关于顺从性和增长间隙的若干经典结果推广到强正递归的可数马尔可夫移位，利用扭曲Patterson--Sullivan测度的方法，并给出扭曲测度的背景及一个例子的估计。
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there ex…
Mixed-identity-freeness and primitivity of group rings
该研究证明了无混合恒等性且含非阿贝尔自由子群的可数群的群环为本原环，提出的动力学准则可复现已有结果并适用于类汤普森群等新群类。
Polynomial filling functions of sublevel sets in $\mathrm{CAT}(0)$ cube complexes
本文给出判据确保 CAT(0) 立方复形高子水平集有多项式积分同调填充函数，并验证 Higman-Thompson 群满足，从而这些群在每度 k≥2 有多项式同调填充函数。
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。
Intersection-saturated groups without free subgroups
本研究提出交饱和群的新判据，据此找到包括Thompson群、Grigorchuk群在内的新实例，证明了无交换自由子群的有限表现交饱和群存在，解决了Delgado等人的问题。
We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.
Stationary boundaries on the space of amenable subgroups and C*-simplicity
本文研究了可数群 G 存在非平凡 μ-边界条件，探讨了 amenable 子群空间的 μ-stationary 性质，并讨论了 Thompson 群 F 的 C*-简单性问题。
Infinite girth and profinite properties of residually finite groups
本文开发两种通用框架构造具有无限围长等性质的剩余有限群，建立标记围长准则为其充要条件，探讨围长与 profinite 完备化的联系并给出线性群的同余实现结果，同时提出相关开放问题。
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