Kazhdan's Property Research Articles

Group cubization

We present a procedure of group cubization: It results in a group whose some features resemble the ones of a given group, and which acts without fixed points on a CAT(0) cubical complex. As a main application we establish lack of Kazhdan's property (T) for Burnside groups.

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 ε such that every unitary representation of G with a (Q,ε)-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=Z, this shows that if (nk)k≥1 is a sequence of integers such that (e2iπθnk)k≥1 is uniformly distributed in the unit circle for all real numbers θ except at most countably many, then {nk;k≥1} is a Kazhdan set in Z as soon as it generates Z. This answers a question of Y. Shalom from [B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T), Cambridge Univ. Press, 2008]. We also obtain characterizations of Kazhdan sets in second countable locally compact abelian groups, in the Heisenberg groups and in the group Aff+(R). This answers in particular a question from [B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T), op. cit.].

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge–Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini–Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces Γ\G/K implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to L 2 (Γ\G). This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral in-variants. On the other hand, when the corresponding Lie group G is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover G/K. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak–Xue. An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups G, we exploit rigidity theory, and in particular the Nevo–Stuck–Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of G.

Property [formula omitted] and property [formula omitted] for Orlicz spaces LΦ

An Orlicz space LΦ(Ω) is a Banach function space defined by using a Young function Φ, which generalizes the Lp spaces. We show, for an Orlicz space LΦ([0,1]) which is not isomorphic to L∞([0,1]), if a locally compact second countable group has property (TLΦ([0,1])), which is a generalization of Kazhdan's property (T) for linear isometric representations on LΦ([0,1]), then it has Kazhdan's property (T). We also show, for a separable complex Orlicz space LΦ(Ω) with gauge norm, Ω=R,[0,1],N, if a locally compact second countable group has Kazhdan's property (T), then it has property (TLΦ(Ω)). We prove, for a finitely generated group Γ and a Banach space B whose modulus of convexity is sufficiently large, if Γ has Kazhdan's property (T), then it has property (FB), which is a fixed point property for affine isometric actions on B. Moreover, we see that, for a hyperbolic group Γ (which may have Kazhdan's property (T)) and an Orlicz sequence space ℓΦΨ with gauge norm such that the Young function Ψ sufficiently rapidly increases near 0, Γ doesn't have property (FℓΦΨ). These results are generalizations of the results for Lp-spaces.

Kazhdan groups whose FC-radical is not virtually abelian

We construct examples of residually finite groups with Kazhdan's property ( T ) whose FC-radical is not virtually abelian. This answer a question of Popa and Vaes about possible fundamental groups of II _1 factors arising from Kazhdan groups.

Property Fℓq implies property Fℓp for 1 < p < q < ∞

It is known that for σ-compact groups Kazhdan's Property (T) is equivalent to Serre's Property FH. Generalized versions of those properties, called properties (TB) and FB, can be defined in terms of the isometric representations of a group on an arbitrary Banach space B. Property FB implies (TB).It is known that a group with Property (Tℓp) shares some properties with Kazhdan's groups, for example compact generation and compact abelianization. Moreover in the case of discrete groups, Property (Tℓp) implies Lubotzky's Property (τ).In this paper we prove that in the case of discrete groups and ℓp(N) spaces, for 1<p<q<∞,p≠2, Property Fℓq implies Property Fℓp.

Embeddings of locally compact hyperbolic groups into Lp-spaces

In the last years, there has been a large amount of research on embeddability properties of finitely generated hyperbolic groups. In this paper, we elaborate on the more general class of locally compact hyperbolic groups. We compute the equivariant Lp-compression in a number of locally compact examples, such as the groups SO(n,1). Next, we show that although there are locally compact, non-discrete hyperbolic groups G with Kazhdan's property (T), it is true that any locally compact hyperbolic group admits a proper affine isometric action on an Lp-space for p larger than the Ahlfors regular conformal dimension of ∂G. This answers a question asked by Yves de Cornulier. Finally, we elaborate on the locally compact version of property A and show that, as in the discrete case, a locally compact second countable group has property A if its non-equivariant compression is greater than 1/2.

On the first cohomology of automorphism groups of graph groups

We study the (virtual) indicability of the automorphism group Aut(AΓ) of the right-angled Artin group AΓ associated to a simplicial graph Γ. First, we identify two conditions – denoted (B1) and (B2) – on Γ which together imply that H1(G,Z)=0 for certain finite-index subgroups G<Aut(AΓ). On the other hand we will show that (B2) is equivalent to the matrix group H=Im(Aut(AΓ)→Aut(H1(AΓ)))<GL(n,Z) not being virtually indicable, and also to H having Kazhdan's property (T). As a consequence, Aut(AΓ) virtually surjects onto Z whenever Γ does not satisfy (B2). In addition, we give an extra property of Γ ensuring that Aut(AΓ) and Out(AΓ) virtually surject onto Z. Finally, in the appendix we offer some remarks on the linearity problem for Aut(AΓ).

Free actions of free groups on countable structures and property (T)

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable !-categorical structures, that is, the closed, oligomorphic permuta- tion groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of G acting freely in all infinite transitive permutation representations of G.

Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

In this series of papers, we study correspondence between the following: (1) large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which appear in the boundary of the set {G(m)}_m in the space of k-marked groups. In this third part of the series, we show the correspondence among the metric properties `geometric property (T),' `cohomological property (T),' and the group property `Kazhdan's property (T).' Geometric property (T) of Willett--Yu is stronger than being expander graphs. Cohomological property (T) is stronger than geometric property (T) for general coarse spaces.

Poincaré inequalities and rigidity for actions on Banach spaces

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group G on a reflexive Banach space X has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that H^1(G,\pi)=0 for every isometric representation \pi of G on X . The condition is expressed in terms of p -Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which H^1(G,\pi) vanishes for every isometric representation \pi on an L_p space for some p&gt;2 . Our methods allow to estimate such a p explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.

Two-spherical topological Kac–Moody groups are Kazhdan

Abstract In this note, we prove that a two-spherical Kac–Moody group over a local field endowed with the Kac–Peterson topology enjoys Kazhdan's property (T).

Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors

In this paper, we give a necessary and sufficient condition for a finitely generated group to have a property like Kazhdan's Property (T) restricted to one isometric representation on a strictly convex Banach space without non-zero invariant vectors. Similarly, we give a necessary and sufficient condition for a finitely generated group to have a property like Property (FH) restricted to the set of the affine isometric actions whose linear part is a given isometric representation on a strictly convex Banach space without non-zero invariant vectors. If the Banach space is the \ell^p space (1&lt;p&lt;\infty) on a finitely generated group, these conditions are regarded as an estimation of the spectrum of the p -Laplace operator on the \ell^p space and on the p -Dirichlet finite space respectively.

On groups with Property [formula omitted

Let 1<p<∞, p≠2. Property (Tℓp) for a second countable locally compact group G is a weak version of Kazhdan Property (T), defined in terms of the orthogonal representations of G on ℓp. Property (Tℓp) is characterized by an isolation property of the trivial representation from the monomial unitary representations of G associated to open subgroups. Connected groups with Property (Tℓp) are the connected groups with a compact abelianization.In the case of a totally disconnected group, isolation of the trivial representation from the quasi-regular representations associated to open subgroups suffices to characterize Property (Tℓp). Groups with Property (Tℓp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of k-regular trees for k≥3 have Property (Tℓp).In the case of discrete groups, Property (Tℓp) implies Lubotzky's Property (τ) and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice Γ in a product G1×G2 of locally compact groups has Property (Tℓp), whenever G1 has Property (T) and G2 is connected and minimally almost periodic. Such a lattice does not have Property (T) if G2 does not have Property (T).

On Kazhdan's Property (T) for the special linear group of holomorphic functions

We investigate when the group $SL_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $SL_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of $SL_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.

On Kazhdan's Property (T) for the special linear group of holomorphic functions

We investigate when the group SLn (O(X)) of holomorphic maps from a Stein space X to SLn (C) has Kazhdan's property (T) for n >= 3. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of SLn (C) has Kazhdan's property (T) for n >= 3. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.

K-theory for the group C*-algebras of a residually finite discrete group with Kazhdan property T

(Z)(n ≥ 3)the n × n special linear groups over the integers. The highly non-trivial and interesting problemto compute the K-theory groups has been considered by the author [6], but it was found to be notcompleted, and to be corrected as [7] (perhaps partly). This time we obtain a sort of solution forthis problem to settle the issue, without using the results of [5] used in [6], but more precisely K

Groups of p-deficiency one

AbstractIn a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

Subspace arrangements and property T

We reformulate and extend the geometric method for proving the 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 on any finite Coxeter group with respect to the standard generating set, which generalizes a result of Bacher and de la Harpe.