Gero Fendler
Let (G,S) be a Coxeter group. We construct a continuation, to the open unit disc, of the unitary representations associated to the positive definite functions g -> r^l(g). (Here 0<r<1, and l denotes the length function with respect to the generating set S.) The constructed representations are uniformly bounded and we prove that this implies the weak amenability of the group G.
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Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,μ)$ such that $L^\infty(X,μ)$ has an invariant mean and that there is a Borel cocycle $α:G\times X\rightarrow H$ which is proper in a suitable, natural sense. We show that if $H$ has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does $G$. We observe that it is the case for a weak form of measure equivalence for pairs of discrete groups.
We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations for which the quantities z^{l(g)} are matrix coefficients. Here l is a length function on G obtained from the combinatorial distance function on the complex X.
An invitation to weak amenability, after Cowling and Haagerup
本文面向局部紧群，系统介绍弱可和性概念，并梳理 Cowling 与 Haagerup 以来的关键研究成果。
Following an approach of Ozawa, we show that several semidirect products are not weakly amenable. As a consequence, we are able to characterize the simply connected Lie groups that are weakly amenable.
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.
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 $\mathbb{Z}\ast H$.
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.
We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an $\tilde{A}_2$-lattice do not have the operator space approximation property and, consequ…
Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{(n)}$ into $A^{(n)}$.
We define the notions of weak amenability and the Cowling-Haagerup constant for extremal finite index subfactors of type II_1. We prove that the Cowling-Haagerup constant only depends on the standard invariant of the subfactor. Hence, we define the Cowling-Haagerup constant for standard invariants. We explicitly compute the constant for Bisch-Haagerup subfactors and prove that it is equal to the constant of the group involved in the construction. Given a finite family of amenable standard invariants in the sense of Popa, we prove that their free product in the sense of Bisch-Jones is weakly am…
In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long s…
We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r^d have uniformly bounded norms for 0<r<1. We then combine this with a Bozejko-Picardello type inequality to obtain weak amenability.
We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. As it turns out, the canonical action of the Coxeter group on its Cayley graph induces a natural action on the compactification and the boundary. From thi…
On weak amenability of Fourier algebras
本文将连通 Lie 群 Fourier 代数弱顺从当且仅当群交换的结论，推广至一般连通局部紧群，扩展了 Choi 和 Ghandehari 的特殊情形方法。
We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin, our result recovers that mapping class groups and Out(F_n) are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann.
A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.
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.
The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda. In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups the…
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 of group actions on compact spaces and the associated Banach algebras
本文将拓扑群顺从性的经典刻画推广至紧 Hausdorff 空间上的离散群作用，通过引入自然关联的 Banach 代数和弱化的顺从性概念，获得了顺从作用的类似刻画，并证明了加强 Dong 和 Wang (2015) 定理的不动点性质。
Extended Weak Order for the Rank 3 Universal Coxeter Group
研究3阶通用考克斯特群的扩展弱序，通过证明戴尔猜想，首次在非球面、非仿射考克斯特群中证实扩展弱序是格，为相关理论研究提供重要成果。
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of…
Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach
提出一种算法计算B型Coxeter群弱序中两个元素的并，将Markowsky算法从标准排列扩展到符号排列，并验证了Dyer关于这些并的几何解释的猜想。
The star operation, originally introduced by Kazhdan and Lusztig, was later specialized by Ernst to the so-called weak star reduction on the set of fully commutative elements of a Coxeter group. In this paper, we classify the star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}_{n+1}$ and $\widetilde{D}_{n+2}$. Focusing then on the case of type $\widetilde{D}_{n+2}$, we use the classification of star irreducible elements to provide a new proof of the faithfulness of a diagrammatic representation of the corresponding generalized Temperley-Li…
On a Conjecture of Dyer on the Join in the Weak Order of a Coxeter group
本文证明了Coxeter群类型A和I的弱序中两个元素join的代数几何描述，并验证了H3和F4类型猜想的正确性。
On the Weak Right Order of a Right-Angled Coxeter System
本文证明在任意Coxeter系统中，所有非单位完全交换元素具有祖先性质，特别地，直角Coxeter系统的所有非单位元素满足该性质，并给出了直角Coxeter系统的公理化刻画。
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.
In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and their position relative to the imaginary cone. Finally, we discuss inversion sets of infinite reduced words and the notions of biconvex and biclosed sets of positive roots.
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…
Affine extended weak order is a lattice
本文证明了仿射类型Coxeter群中扩展弱序是一个完全格，通过引入清洁排列的概念并建立通用框架进行验证。
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.
Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz the relative weak topology of C(X,M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology.
We prove that the weak order on an infinite Coxeter group contains infinite antichains if and only if the group is not affine.
We show that every finitely generated Artin-Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.
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。
In this article we provide a new finite class of elements in any Coxeter system (W,S) called low elements. They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W,S). Our first main result is to show that they form a Garside shadow in (W,S), i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated Artin-Tits groups have a finite Garside family. This result was announced in a note with P. Dehornoy (P. Dehornoy, M. Dyer, and C. Hoh…
Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible word…
Von Neumann equivalence and $M_d$ type approximation properties
本文研究$M_d$型逼近性质，证明三类性质在von Neumann等价（含测度等价与W*-等价）下稳定，并可由格继承。
Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point ξ, a natural order-preserving correspondence between infinite reduced words which "point towards" ξ, and elements of the ref…
Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s \in S$. If there exists $t\in S$ such that $s$ and $t$ do not commute and $tw$ (respectively, $wt$) is no longer fully commutative, we say that $w$ is left (respectively, right) weak star reducible by $s$ with respect to $t$. In this paper, we classify the fully commutative elements in Coxeter groups of types $B$ and affine $C$ that are irreducible under weak star reductions. In a sequel to this paper, the classificat…
In this paper we consider the class of connected simple Lie groups equipped with the discrete topology. We show that within this class of groups the following approximation properties are equivalent: (1) the Haagerup property; (2) weak amenability; (3) the weak Haagerup property. In order to obtain the above result we prove that the discrete group GL(2,K) is weakly amenable with constant 1 for any field K.
We obtain vanishing ranges for the mod $p$ cohomology of alternating subgroups of finite $p$-free Coxeter groups. Here a Coxeter group $W$ is $p$-free if the order of the product $st$ is prime to $p$ for every pair of Coxeter generators $s,t$ of $W$. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite $p$-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxet…
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 …
The $\widetilde{A}_n$ Coxeter groups are known to not be systolic or cocompactly cubulated for $n\geq 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity we describe the canonical emeddings of the 1-skeleta of $\widetilde{A}_n$ Coxeter complexes into the Euclidean spaces $\mathbb{R}^{n+1}$. We also prove weak modularity for buildings of type $\widetilde{A}_3$.
The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto's theorem. Therefore the word problem is solved for the groupoid.
Weak order on groups generated by involutions
本文研究了involution系统中的弱序结构，探讨了其作为完全meet-semilattice的条件，并讨论了相关性质及开放问题。
The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule a…
Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups
本文研究了Coxeter群中元素的规范连接表示和连接不可约元素的性质，并提出了确定元素锥型的高效计算方法。
This is the first of a series of papers which define and study structures called rootoids, which are groupoids equipped with a representation in the category of Boolean rings and with an associated 1-cocycle. The axioms for rootoids are abstracted from formal properties of Coxeter groups with their root systems and weak orders. They imply that each of the weak orders of a rootoid embeds as an order ideal in a complete ortholattice. This first paper is concerned only with the most basic definitions, facts and examples; the main results, which are new even for Coxeter groups, will be stated and …
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…
Let $(W,S)$ be an arbitrary Coxeter system. For each word $ω$ in the generators we define a partial order--called the {\sf $ω$-sorting order}--on the set of group elements $W_ω\subseteq W$ that occur as subwords of $ω$. We show that the $ω$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the $ω$-sorting order is a "maximal lattice" in the sense that the addition of any collection of Bruhat covers results in a nonlattice. Along the way we define a class of structures called {\sf supersolvable antimatro…
Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity
本文利用双模系数的 Schatten 类条件刻画弱包含关系，并研究 Coxeter 群梯度双模，建立 gradient-S_p、紧化小性与奇偶路径的等价性，从而推广强刚性结果。
For an infinite Coxeter system, one can extend the weak right order to the set of infinite reduced words. This is called limit weak order. In [Transformation Groups 18(1), 2013, 179-231], Lam and Pylyavskyy showed that for affine Weyl groups of type $\widetilde{A}_n$ minimal infinite reduced words under the limit weak order are precisely those infinite Coxeter elements and asked the question of characterization, in terms of infinite reduced words, of the minimal elements of the limit weak order for other affine types. In this paper, we answer this question by characterizing the minimal infinit…
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…
Classification of weak Bruhat interval modules of $0$-Hecke algebras
本文以类型无关的方式给出了 Coxeter 群中两个左弱 Bruhat 区间保降序同构的等价描述，并分类了 $0$-Hecke 代数的所有弱 Bruhat 区间模，解决了 Jung-Kim-Lee-Oh 的公开问题并证实了 Kim-Lee-Oh 的猜想。
In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W,S). The definition of these automata is straightforward as it only requires the notion of weak order on (W,S) and the related notion of Garside shadows in (W,S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) …
A structure of a complete lattice (in the sense of a poset) is defined on the underlying set of the orhtogonal group of a real Euclidean space, by a construction analogous to that of the weak order of a Coxeter system in terms of its root system. This gives rise to a complte rootoid in the sense of Dyer, with the orthogonal group as underlying group.
We introduce the notion of weakly systolic complexes and groups, and initiate regular studies of them. Those are simplicial complexes with nonpositive-curvature-like properties and groups acting on them geometrically. We characterize weakly systolic complexes as simply connected simplicial complexes satisfying some local combinatorial conditions. We provide several classes of examples --- in particular systolic groups and CAT(-1) cubical groups are weakly systolic. We present applications of the theory, concerning Gromov hyperbolic groups, Coxeter groups and systolic groups.
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