Uniform Kazhdan groups

Kazhdan quotients of Golod-Shafarevich groups

The main goal of this paper is to prove that every Golod-Shafarevich group has an infinite quotient with Kazhdan's property $(T)$. In particular, this gives an affirmative answer to the well-known question about non-amenability of Golod-Shafarevich groups.

Uniform Kazhdan Constant for some families of linear groups

Let $R$ be a ring generated by $l$ elements with stable range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant $ε_0>0$ for some $d > r $. We prove that there exist $ε(ε_0,l) >0$ and $k \in N$, s.t. for every $n \geq d$, $EL_n(R)$ has a generating set of order $k$ and a Kazhdan constant larger than $ε$. As a consequence, we obtain for $SL_n(Z)$ where $n \geq 3$, a Kazhdan constant which is independent of $n$ w.r.t generating set of a fixed size.

A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.

线性群小生成集的小Kazhdan常数

Small Kazhdan constants for small generating sets of linear groups

本文证明线性群及具有非初等收敛作用的有限生成群，存在大小仅比生成元数多2的生成集，其Kazhdan常数可任意小。

具有卡扎丹基本群的余维数为二的流形中的一致腰不等式

Uniform waist inequalities in codimension two for manifolds with Kazhdan fundamental group

研究具有卡扎丹基本群的闭黎曼流形M，已知其有限覆盖在余维数一时满足一致腰不等式，现证明其有限覆盖在余维数二时同样满足该不等式。

$n$-Kazhdan groups and higher spectral expanders

Let $Γ$ be a group of type $F_n$ and let $X$ be the $n$ skeleton of the universal cover of a $K(Γ,1)$ simplicial complex with finite $n$ skeleton. We show that if $Γ$ is strongly $n$-Kazhdan, then for any family of finite index subgroups $\{Λ_i\}_i$, the family of simplicial complexes $\{Λ_i\backslash X\}_i$ are bounded degree $n$-dimensional spectral expanders. Using this we construct new examples of $2$ dimensional spectral expanders.

Kazhdan groups with infinite outer automorphism group

For each countable group $Q$ we produce a short exact sequence $1\to N \to G \to Q\to 1$ where $G$ is f.g. and has a graphical $\frac16$ presentation and $N$ is f.g. and satisfies property $T$. As a consequence we produce a group $N$ with property $T$ such that $\Out(N)$ is infinite. Using the tools developed we are also able to produce examples of nonHopfian and non-coHopfian groups with property $T$. One of our main tools is the use of random groups to achieve certain properties.

Kazhdan-Margulis theorem for Invariant Random Subgroups

Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every $ε$ there is an identity neighbourhood $U$ which intersects trivially the stabilizers of $1-ε$ of the points in every non-atomic $G$-space.

Conjugating automorphisms of graph products: Kazhdan's property (T) and SQ-universality

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan's property (T) and when it satisfies some vastness properties including SQ-universality.

Rips construction and Kazhdan property (T)

We show that for any non--elementary hyperbolic group $H$ and any finitely presented group $Q$, there exists a short exact sequence $1\to N\to G\to Q\to 1$, where $G$ is a hyperbolic group and $N$ is a quotient group of $H$. As an application we construct a hyperbolic group that has the same $n$--dimensional complex representations as a given finitely generated group, show that adding relations of the form $x^n=1$ to a presentation of a hyperbolic group may drastically change the group even in case $n>> 1$, and prove that some properties (e.g. properties (T) and FA) are not recursively recogni…

Kazhdan constants of group extensions

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of $p$-groups of lower $p$-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

Coarse median structures and homomorphisms from Kazhdan groups

We study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan's property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock-Drutu-Sapir on the mapping class groups of ori…

Relative Kazhdan Property

We perform a systematic investigation of Kazhdan's relative Property (T) for pairs (G,X), where G a locally compact group and X is any subset. When G is a connected Lie group or a p-adic algebraic group, we provide an explicit characterization of subsets X of G such that (G,X) has relative Property (T). In order to extend this characterization to lattices of G, a notion of "resolutions" is introduced, and various characterizations of it are given. Special attention is paid to subgroups of SU(2,1) and SO(4,1).

The equivariant Kazhdan-Lusztig polynomial of a matroid

We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

New Kazhdan groups

The group of simplicial automorphisms of a Tits-Kac-Moody ininite building of thickness q associated to a cocompact reflexion group with fundamental domain a simplex, is Kazhdan for q sufficiently large. Thus we obtain families of new Kazhdan groups: two in dimension 3 and one in dimension 4. The proof uses continuos cohomology, in particular a lemma of Borel-Wallach, and Garland's vanishing method.

小取消条件与具有 (T) 性质的超限群在外自同构群中的作用

Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces

该研究证明了每个有限群都可以作为具有 (T) 性质的超限群的外自同构群，并展示了如何构造具有特定外自同构群的超限群。

Kazhdan群的拟对角性

Quasidiagonality of Kazhdan Groups

本研究针对Lubotzky和Shalom的猜想，证明无限离散Kazhdan性质(T)群不具有有限维密度(FDD)性质，肯定解决了该猜想。

高阶Brin--Thompson群具有Kazhdan性质

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。

Mapping Class Groups do not have Kazhdan's Property (T)

We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T).

Kazhdan-Lusztig cells in the affine Weyl groups of rank 2

In this paper we determine the partition into Kazhdan-Lusztig cells of the affine Weyl groups of type $\tB_{2}$ and $\tG_{2}$ for any choice of parameters. Using these partitions we show that the semicontinuity conjecture of Bonnafé holds for these groups.

Profinite completions and Kazhdan's property (T)

We show that property (T) is not profinite, that is, we construct two finitely generated residually finite groups which have isomorphic profinite completions while one admits property (T) and the other does not. This settles a question raised by M. Kassabov.

Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig. Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases. Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conject…

On fixed point property for $L_p$-representations of Kazhdan groups

Let $G$ be a topological group with finite Kazhdan set, let $Ω$ be a standard Borel space and $μ$ a finite measure on $Ω$. We prove that for any $p\in [1, \infty)$, any affine isometric action $G \curvearrowright L_p(Ω, μ)$ whose linear part arises from an ergodic measure-preserving action $G \curvearrowright (Ω, μ)$, has a fixed point.

Relative inner amenability, relative property gamma and non-Kazhdan groups

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).

非Kazhdan群与trace von Neumann代数的一种新近似性质

Non Kazhdan's groups and a new approximation property for tracial von Neumann algebras

本文定义了trace von Neumann代数的新近似性质，该性质在离散群和II₁因子中等价于Kazhdan性质(T)的否定。

Kazhdan groups have cost 1

We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group $\mathbb{Z}$

The aim of this partly expository paper is to present and discuss two classes of sets of integers (Jamison and Kazhdan sets) whose definition and/or properties are determined or inspired by operator-theoretical properties. Jamison sets first appeared in the study of the relationship between the growth of the sequence of norms of iterates of a bounded linear operator on a separable Banach space and the size of its unimodular point spectrum. Kazhdan subsets of $\mathbb{Z}$ are particular cases of Kazhdan sets in general topological groups, which are especially important as they appear in the def…

On complements of Kazhdan projections in semisimple groups

We prove that for an isometric representation of some groups on certain Banach spaces, the complement of the subspace of invariant vectors is 1-complemented.

Semi-continuité des cellules de Kazhdan-Lusztig

Computations in small Coxeter groups and infinite dihedral groups suggest that Kazhdan-Lusztig cells for unequal parameters obey to some "semicontinuity" phenomenon (as the parameter vary). The aim of this paper is to provide a rigorous theoretical background that allows to state some precise conjectures.

整数环上Chevalley群的Kazhdan常数

Kazhdan constants for Chevalley groups over the integers

本文研究整数环上Chevalley群的Kazhdan常数下界问题，通过建立根系结构与Laplace算子平方行为的新联系，得到了除$\u27e8\mathtt{A}_{n}\u27e9$外类型的首个显式渐近精确下界。

Computing Kazhdan constants by semidefinite programming

Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of a discrete group. Positive lower bounds imply that the group has property (T). We study lattices on $\tilde{A}_2$-buildings in detail. For $\tilde{A}_2$-groups, our numerical bounds look identical to the known actual constants. That suggests that our approach is effective. For a family of groups, $G_1, \cdots, G_4$, that are studied by Ronan, Tits and others…

维度为16的Kazhdan群及其预设的第二个ℓ²-贝蒂数

Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number

研究构建了具有性质(T)的简单稀疏超几何群，其理性上同调维数为16，且第二个ℓ²-贝蒂数可预设为任意正实数。同时，还构建了具有性质(T)的超几何群，其第二个ℓ²-贝蒂数可预设为任意非负有理数。

A characterization of relative Kazhdan Property T for semidirect products with abelian groups

Let A be a locally compact abelian group, and H a locally compact group acting on A. Let G=HA be the semidirect product. We prove that the pair (G,A) has Kazhdan's Property T if and only if the only countably approximable H-invariant mean on the Borel subsets of the Pontryagin dual of A, supported at the neighbourhood of the trivial character, is the Dirac measure.

Kazhdan sets in groups and equidistribution properties

Using functional and harmonic analysis methods, we study Kazhdan sets in topological groups which do not necessarily have Property (T). We provide a new criterion for a generating subset $Q$ of a group $G$ to be a Kazhdan set; it relies on the existence of a positive number $\varepsilon$ such that every unitary representation of $G$ with a $(Q,\varepsilon )$-invariant vector has a finite dimensional subrepresentation. Using this result, we give an equidistribution criterion for a generating subset of $G$ to be a Kazhdan set. In the case where $G=\mathbb{Z}$, this shows that if $(n_{k})_{k\ge 1…

Amenability versus property (T) for non locally compact topological groups

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 …

作用的不可近似性与Kazhdan性质(T)

Inapproximability of actions and Kazhdan's property (T)

该研究针对具有Kazhdan性质(T)的可数群G与有限生成群D，通过sofic逼近的扩张子图性质，推导D的LEF性，明确非LEF时G×D作用的极限性质，并给出几乎自同构的自改进定理作为关键技术支撑。

一个非一致指数增长的圆柱双曲群

An acylindrically hyperbolic group of non-uniform exponential growth

构造了一个具有非一致指数增长和Kazhdan性质(T)的圆柱双曲群，该群是递增有限秩自由群的自同构群的公共商。

算术群的更高Kazhdan性质与单位元余维数

Higher Kazhdan property and unitary cohomology of arithmetic groups

本文研究了算术群的更高Kazhdan性质，通过改进稳定性范围并允许任意单位元表示，推广了Borel和Borel-Yang的定理，并引入了几何群论和Clozel的工作方法。

Rigidity sequences, Kazhdan sets and group topologies on the integers

We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences wh…

An alternative proof of Kazhdan property for elementary groups

In 2010, Invent. Math., Ershov and Jaikin-Zapirain proved Kazhdan's property (T) for elementary groups. This expository article focuses on presenting an alternative simpler proof of that. Unlike the original one, our proof supplies no estimate of Kazhdan constants. It may be regarded as a specific example of the results in the paper "Upgrading fixed points without bounded generation" (arXiv:1505.06728, forthcoming version) by the author.

Local rigidity for actions of Kazhdan groups on non commutative $L_p$-spaces

Given a discrete group $Γ$, a finite factor $\mathcal N$ and a real number $p\in [1, +\infty)$ with $p\neq 2,$ we are concerned with the rigidity of actions of $Γ$ by linear isometries on the $L_p$-spaces $L_p(\mathcal N)$ associated to $\mathcal N$. More precisely, we show that, when $Γ$ and $\mathcal N$ have both Property (T) and under some natural ergodicity condition, such an action $π$ is locally rigid in the group $G$ of linear isometries of $L_p(\mathcal N)$, that is, every sufficiently small perturbation of $π$ is conjugate to $π$ under $G$. As a consequence, when $Γ$ is an ICC Kazhdan…

Subspace Arrangements and Property T

We reformulate and extend the geometric method for proving Kazhdan property T developed by Dymara and Januszkiewicz and used by Ershov and Jaikin. The main result says that a group G, generated by finite subgroups G_i, has property T if the group generated by each pair of subgroups has property T and sufficiently large Kazhdan constant. Essentially, the same result was proven by Dymara and Januszkiewicz, however our bound for "sufficiently large" is significantly better. As an application of this result, we give exact bounds for the Kazhdan constants and the spectral gaps of the random walks o…

Semicontinuity properties of Kazhdan-Lusztig cells

Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures.

Lipschitz 和 Kazhdan 常数之间的一个联系：关于实数线上同胚群的群

A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line

研究了实数线上同胚群的Lipschitz和Kazhdan常数之间的联系，得出了相关群作用的Lipschitz和Kazhdan常数的上下界。

The structure of the nilpotent cone, the Kazhdan-Lusztig map and algebraic group analogues of the Slodowy slices

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.

混合积群的高阶Kazhdan投影和离域ℓ²-Betti数

Higher Kazhdan projections and delocalized $\ell^2$-Betti numbers for an amalgamated product group

本文为混合积群ℤₘ*_(ℤ_d)ℤₙ的高阶Kazhdan投影建立K-理论类显式表达式，推广了Pooya和Wang关于自由积群的结果，并应用该表达式得到SL(2,ℤ)离域ℓ²-Betti数的非零结果。

Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum-Connes conjectures

We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.

L_p空间的子空间与商空间的Kazhdan性质(T)

Kazhdan's Property $(T)$ for Subspaces and Quotients of $L_p$-Spaces

该研究针对实L_p空间的闭子空间与商空间，证明了等距表示的一致谱间隙估计，结合已有工作肯定回答了Kazhdan性质(T)蕴含对应性质(T_X)的问题。

高阶Kazhdan投影与离域$\ell^2$-Betti数

Higher Kazhdan projections and delocalised $\ell^ 2$-Betti numbers

本文对特定自由积群和直积群的高阶Kazhdan投影的K-类给出了显式描述，并利用该描述计算了Lott的离域$\ell^2$-Betti数，首次得到了无限群的非零结果。

A local characterization of Kazhdan projections and applications

We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent for $σ$-compact locally compact groups. Another kind of applications…

On the Haagerup and Kazhdan properties of R. Thompson's groups

A machine developed by the second author produces a rich family of unitary representations of the Thompson groups F,T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero [F,F]-invariant vectors reproving, at least for T, Reznikov's result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan's property (T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibi…

Noncommutative real algebraic geometry of Kazhdan's property (T)

It is well-known that a finitely generated group $Γ$ has Kazhdan's property (T) if and only if the Laplacian element $Δ$ in ${\mathbb R}[Γ]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ${\mathbb R}[Γ]$. Namely, $Γ$ has property (T) if and only if there are a constant $κ>0$ and a finite sequence $ξ_1,...,ξ_n$ in ${\mathbb R}[Γ]$ such that $Δ^2-κΔ= \sum_i ξ_i^*ξ_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in ${\mathbb R}[Γ]$, possibly with an assist of computers.

几何性质 (T) 与 Kazhdan 投影

Geometric Property (T) and Kazhdan projections

本文利用最大一致 Roe 代数中的投影刻画几何性质 (T)，将群的 Kazhdan 投影推广至度量空间，并通过粗连通分支分解描述该投影。

Kazhdan projections, random walks and ergodic theorems

In this paper we investigate generalizations of Kazhdan's property $(T)$ to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. They exhibit useful properties and flexibility, and allow to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address …