E. T. Shavgulidze
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.
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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.
We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem. We study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X. Since relative amenability is closed under Chabauty limits, it follows that any Chabauty l…
Amenability and skew-amenability of actions of topological groups
本文定义并研究拓扑群在紧空间上连续作用的可均性与斜可均性，旨在解决可均性向闭子群传递的条件，并应用于泛最小流分析。
We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.
On invariant means and pre-syndetic subgroups
本文研究非局部紧拓扑群顺从性变体，证明斜顺从性可被预共生子群继承，并探讨 Malliavin 提出的顺从性变体，建立其 Reiter 型和 Følner 型判据。
This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.
Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amena…
We adapt the notion of quasi-Hermition group to the pairs $(G,H)$ of discrete group $G$ and its subgroup $H$. We show that a quasi-Hermitian pair is amenable in the sense of Eymard.
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 …
We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.
Closed approximate subgroups: compactness, amenability and approximate lattices
本文研究了局部紧群的闭近似子群与近似格，利用好模型概念证明了 Cartan 闭子群定理的近似版本及顺从群的结构定理，并推广了 Meyer 准晶结构定理与 Auslander-Bieberbach-Mostow 定理。
Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and the cube of u is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda and formation of diagonal products, we construct an example for which the latter dense group is non-exact. Our proof employs approximations in the space of marked groups of LEF ("Locally Embeddable into…
We first show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address co-amenability for von Neumann algebras, describing notably how it relates to the former.
We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann -- Day problem concerning amenability of groups without non--abelian free subgroups. As another consequence, we obtain that the above--mentioned groups are of uniform exponential growth.
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…
In this short paper we show that if $(G,H)$ is an amenable Hecke pair and $Γ$ is a subgroup of $G$ containing $H$, then the Hecke pair $(Γ, H)$ is amenable too. This answers positively a question of Pierre Eymard, asked in 1972, when it is restricted to Hecke pairs
P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.
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.
We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups, each of which has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with at least one stabilizer of property A. Specializing the…
Let $G$ be a subgroup of a discrete (countable) group $Γ$. We introduce a notion of relative inner amenability of $G$ in $Γ$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss briefly the corresponding relative property gamma for pairs of finite factors and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property $T$.
We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable.
Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of finite factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property (T).
We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of M.
Let O(f,Z) be the integral orthogonal group of an integral quadratic form f of signature (n,1). Let R(f,Z) be the subgroup of O(f,Z) generated by all hyperbolic reflections. Vinberg proved that if n > 29 then the reflective quotient O(f,Z)/R(f,Z) is infinite. In this note we generalize Vinberg's theorem and prove that if n > 91 then O(f,Z)/R(f,Z) contains a non-abelian free group (and thus it is not amenable).
We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.
We define a notion of relative soficity for countable groups with respect to a family of groups. A group is sofic if and only if it is relative sofic with respect to the family consisting only of the trivial group. If a group is relatively sofic with respect to a family of sofic groups, then the group is sofic. Using this notion we generalize a theorem of Elek and Szabo on the soficity of an extension of sofic groups by amenable groups. In particular we prove that groups that are relatively sofic with respect to a family of sofic groups are sofic and more importantly extensions of sofic groups…
We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbolΛ(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the con…
We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.
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.
We prove that every right-angled Artin group embeds into the $C^{\infty}$ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the $C^{\infty}$ diffeomorphism group of the real line.
We show that certain amenable subgroups inside $\tilde{A}_2$-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann subalgebras. We also give a geometric proof that if $G$ is an acylindrically hyperbolic group, $H$ is an infinite amenable subgroup containing a loxodromic element, then $H<G$ is singular. Finally, we present (counter)examples to show both situations happen concerning maximal amenability of $LH$ inside $LG$ if $H$ does not contain loxodromic elements.
We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.
Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\R$, under the operation of composition, and let $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \Diffeo$ are conjugate in $\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to t…
We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjec…
Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot …
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples includ…
The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. There are proved theorems about projective limit decompositions of these groups and their compactifications for compact manifo…
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 idempote…
We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.
We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.
Bounded cohomology of classifying spaces for families of subgroups
本文引入子群族的有界 Bredon 上同调，推广有界上同调，并给出相对可遗性与相对双曲性的上同调刻画。
Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not…
Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces over non-Archimedean fields $\bf F$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex manifolds. It is proved that $G$ is not a Banach-Lie group, but it has a neighbourhood $W$ of the unit element $e$ such that each element $g$ in $W$ belongs to at least one corresponding one-parameter subgroup. It is proved that $G$ is simple and perfect. Its compact subgroups $G_c$ are studied such that a dimension over $\bf F$ of its tangent space $dim_{\bf F}T…
On inner-amenability and boundary actions
研究离散可数群\(\Gamma\)，证明若\(\Gamma\)是 ICC 内顺从非顺从的则不满足(AO)性质，推广该现象；还证明若\(\Gamma\)是特定离散子群则不可能内顺从，推广了相关已知事实。
Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely ge…
We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.
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 …
An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.
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。
The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.
Boundary values of diffeomorphisms of simple polytopes, and controllability
本文研究简单多胞形上微分同胚群的边界值问题，建立了商空间的典范李群结构，并证明恒等分量由指数映射生成，得出可控性结论。
We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms.
In 'The essentially chief series of a compactly generated locally compact group', an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors necessarily exist in all locally compact groups with sufficiently rich topological structure. We also show that chief factors have one of seven types, and for all but one of these types, there is a decomposition into discrete groups, compact groups, and topologically simple groups. Our results for chief factors require exploring the theory developed in 'Chi…
Stochastic processes are considered on free loop spaces, geometric loop and diffeomorphism groups of real and complex manifolds. They are used for investigations of Wiener differentiable quasi-invariant measures on such groups relative to dense subgroups. There are considered finite-dimensional and also infinite-dimensional manifolds. Such measures are used for the investigation of associated unitary representations of these groups.
We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field $K$. To capture the features of classical amenability that induce the vanishing of real bounded cohomology, we introduce the notion of normed $K$-amenability, of which we prove an algebraic characterization. It implies that normed $K$-amenable groups are locally elliptic, and it relates an invariant, the norm of a $K$-amenable group, to the order of its discrete finite $p$-subquotients, where $p$ is the characteristic of the residue field of $K$. Moreover, w…
A family of exotic group C*-algebras
研究不含非阿贝尔自由子群的群的非顺从性强化性质，通过显式构造证明这类群拥有丰富的可定义奇特C*-代数的不可约表示，丰富了群C*-代数相关研究。
Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the prope…
The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space. We prove that any finitely generated subgroup of $\mathrm{Aff}(S,q)$ is undistorted in the mapping class group. We also show that the systole map restricted to the associated electrified Nielsen core in the Teichmüller disc is a quasi-isometric embedding into the curve complex.
On groups of diffeomorphisms of the interval with finitely many fixed points
本文针对每个非单位元仅有有限多个不动点的区间保向微分同胚群，证明其不可约子群要么为仿射群，要么具有无限围道，从而肯定了Girth Alternative。
Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction
本文针对实解析圆周微分同胚群的局部离散几乎自由子群，通过推广乒乓引理给出拓扑动力学分类方法，并在受限情形下解决了Dippolito的叶状结构理论古老猜想，相关推论将在配套论文详述。
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