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 recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier--Wise on outer automorphism groups of Kazhdan groups.
We generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's construction of torsion-free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product. Our graphical small cancellation theorem then provides first examples of Gromov hyperbolic groups without the unique product property. We construct uncountably many non-isomorphic torsion-free…
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.
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.
We show that the standard set of elementary generators of an elementary isotropic reductive group over a connected finitely generated ring is a Kazhdan subset. This generalizes the corresponding result of M. Ershov, A. Jaikin-Zapirain, and M. Kassabov for Chevalley and twisted Chevalley groups.
We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.
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).
We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.
An Orlicz space $L^Φ(Ω)$ is a Banach function space defined by using a Young function $Φ$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^Φ([0,1])$, a locally compact second countable group has Kazhdan's property $(T)$ if and only if it has property $(T_{L^Φ([0,1])})$, which is a generalization of Kazhdan's property $(T)$ for linear isometric representations on $L^Φ([0,1])$. We also prove that, for a Banach space $B$ whose modulus of convexity is sufficiently large, if a locally compact second countable group has Kazhdan's property $(T)$, then it has property…
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.
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 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…
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.
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.
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…
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 …
Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for ${\rm SL}(3,\mathbb Z)$, by showing that $Δ^2- \frac{1}{6} Δ$ is a hermitian sum of squares in the group algebra, where $Δ$ is the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of $\frac{1}{72}\sim 0.014$ for the associated random walk operator. The sum of squares representation was found numerically by a semidefinite programming algorithm, and then turned into an exact symbolic representation, provided in an attached Mathematica file.
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.
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.
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.
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitiv…
We show that a group with Kazhdan's property $(T)$ has property $(T_{B})$ for $B$ the Haagerup non-commutative $L_{p}(\mathcal{M})$-space associated with a von Neumann algebra $\mathcal{M}$, $1
Given a Lie algebra \s, we call Lie \s-algebra a Lie algebra endowed with a reductive action of \s. We characterize the minimal \s-Lie algebras with a nontrivial action of \s, in terms of irreducible representations of \s and invariant alternating forms. As a first application, we show that if \g is a Lie algebra over a field of characteristic zero whose amenable radical is not a direct factor, then \g contains a subalgebra which is isomorphic to the semidirect product of sl_2 by either a nontrivial irreducible representation or a Heisenberg group (this was essentially due to Cowling, Dorofaef…
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…
Let p be a real number with 1<p and different from 2. We study Property (T_lp) for a second countable locally compact group G. Property (T_lp) is a weak version of Kazhdan's Property (T), defined in terms of the orthogonal representations of G on the sequence space lp. We show that Property (T_lp) for a totally disconnected group G is characterized by an isolation property of the trivial representation from the quasi-regular representations associated to open subgroups of G. Groups with Property (T_lp) share some important properties with Kazhdan groups (compact generation, compact abelianizat…
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…
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.
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…
Property FW is a natural combinatorial weakening of Kazhdan's Property T. We prove that the group of piecewise homographic self-transformations of the real projective line, has "few" infinite subgroups with Property FW. In particular, no such subgroup is amenable or has Kazhdan's Property T. These results are extracted from a longer paper. We provide a complete proof, whose main tools are the use of the notion of partial action and of one-dimensional geometric structures.
We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric gr…
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 $(T_{B})$ and $(F_{B})$, can be defined in terms of the isometric representations of a group on an arbitrary Banach space $B$. Property $(F_{B})$ implies $(T_{B})$. It is known that a group with Property $(T_{l_p})$ shares some properties with Kazhdan's groups, for example compact generation and compact abelianization. Moreover in the case of discrete groups, Property $(T_{l_p})$ implies Lubotzky's Property $(τ)$. In this paper w…
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 …
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…
We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger.
The aim of the article is to provide a characterization of Kazhdan's property (T) for locally compact, second countable pairs of groups $H\subset G$ in terms of actions on infinite, $σ$-finite measure spaces. It is inspired by the recent characterization of the Haagerup property by similar actions due to T. Delabie, A. Zumbrunnen and the author.
We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relative…
We prove that $\operatorname{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.
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 $ω$-categorical structures, that is, the closed, oligomorphic permutation 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…
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…
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…
