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, Dorofaeff, Seeger, and Wright). As a corollary, if G is an algebraic group over a local field K of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then G(K) has Property (T) relative to a noncompact subgroup. In particular, G(K) does not have Haagerup's property. This extends a similar result of Cherix, Cowling and Valette for connected Lie groups, to which our method also applies.
We give some other applications. We provide a characterization of connected Lie groups all of whose countable subgroups have Haagerup's property. We give an example of an arithmetic lattice in a connected Lie group which does not have Haagerup's property, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise non-isomorphic connected Lie groups with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.
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.
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 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
We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath pr…
The Haagerup property, which is a strong converse of Kazhdan's property $(T)$, has translations and applications in various fields of mathematics such as representation theory, harmonic analysis, operator K-theory and so on. Moreover, this group property implies the Baum-Connes conjecture and related Novikov conjecture. The Haagerup property is not preserved under arbitrary group extensions and amalgamated free products over infinite groups, but it is preserved under wreath products and amalgamated free products over finite groups. In this paper, we show that it is also preserved under graph p…
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 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).
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.
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.
We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the $L^p$-versions of Property (FH) and a-(FH)-menability as well as quasified versions thereof a la Burger--Monod and Ozawa.
This habilitation memoir (in French, submitted in May 2014) is made up of five chapters, each being an introduction to work of the author between 2006 and 2014. The core of the memoir consists of the first three chapters, pertaining to geometric group theory. More precisely, it is concerned with large-scale geometry of groups and especially Lie groups (Chapters 1 and 2), and locally compact hyperbolic groups (Chapter 3). Chapter 4 is concerned with the study of groups through their unitary representations and their isometric actions on Hilbert spaces (Kazhdan and Haagerup properties). Finally,…
In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_ζ$ (at roots of unity $ζ$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $ζ$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $Φ$. For …
Given a finite root system $Φ$, we show that there is an integer $c=c(Φ)$ such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with root system $Φ$ and any irreducible rational $G$-modules $L,L'$. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for $\Ext^n$, for any integer $n\geq 0$, using a constant depending only on $n$ and the root system. Weaker versions of this are proved in the algebraic and finite group cases, sufficient to give similar r…
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…
We prove the finiteness of formal analogues of the spherical function (Spherical Finiteness), the ${\mathbf c}$-function (Gindikin-Karpelevich Finiteness), and obtain a formal analogue of Harish-Chandra's limit (Approximation Theorem) relating spherical and ${\mathbf c}$-function in the setting of $p$-adic Kac-Moody groups. The finiteness theorems imply that the formal analogue of the Gindikin-Karpelevich integral is well defined in local Kac-Moody settings. These results extend the Braverman-Garland-Kazhdan-Patnaik's affine Gindikin-Karpelevich finiteness theorems and provide an algebraic ana…
Let $G$ be the symplectic group $Sp_4$ over a non Archimedean local field of any characteristic. It is proved in this paper that for $p\in[1,4/3)\cup (4,\infty]$ neither the group $G$ nor its lattices have the property of approximation by Schur multipliers on Schatten $p$ class ($AP_{pcb}^{Schur}$) of Lafforgue and de la Salle. As a consequence, for any lattice $Γ$ in $G,$ the associated non-commutative $L^p$ space $L^p(LΓ)$ of its von Neumann algebra $L(Γ)$ fails the operator space approximation property (OAP) and completely bounded approximation property (CBAP) for $p\in[1,4/3)\cup (4,\infty…
Generally speaking, this thesis focuses on the interplay between the representations of Lie groups and probability theory. It subdivides into essentially three parts. In a first rather algebraic part, we construct a path model for geometric crystals in the sense of Berenstein and Kazhdan, for complex semi-simple Lie groups. We will mainly describe the algebraic structure, its natural morphisms and parameterizations. The theory of total positivity will play a particularly important role. Then, we anticipate on the probabilistic part by exhibiting a canonical measure on geometric crystals. It us…
