Higher-rank Lie Groups Research Articles

Bounded cohomology is not a profinite invariant

AbstractWe construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher-rank simple Lie groups. Using Galois cohomology, we actually show that $\operatorname {SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups $E_{6(-14)}$ and $E_{7(-25)}$ constitute the complete list of higher-rank Lie groups admitting such examples.

On absolutely profinitely solitary lattices in higher rank Lie groups

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp ⁡ ( n , R ) \operatorname {Sp}(n,\mathbb {R}) , G 2 ( 2 ) G_{2(2)} , E 8 ( C ) E_8(\mathbb {C}) , F 4 ( C ) F_4(\mathbb {C}) , and G 2 ( C ) G_2(\mathbb {C}) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

On the profinite rigidity of lattices in higher rank Lie groups

AbstractWe investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$ , $F_4$ , and $G_2$ . In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\textrm{SL}_{2n+1}(\mathbb{R})$ , $\textrm{SL}_{2n+1}(\mathbb{C})$ , $\textrm{SL}_n(\mathbb{H})$ , or groups of type $E_6$ .

Stable random fields, Patterson–Sullivan measures and extremal cocycle growth

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, and (3) G is the mapping class group of a surface acting on its Teichmüller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson–Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen–Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric \(\alpha \)-stable (\(0< \alpha < 2\)) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.

Adelic superrigidity and profinitely solitary lattices

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher-rank Lie groups

Let \(\mathbb{G}\) denote a higher-rank ℝ-split simple Lie group of the following type: SL(n, ℝ), SOo(m, m), E6(6), E7(7) and E8(8), where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete, abelian parabolic actions on \(\mathbb{G}\) via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case \(\mathbb{G}\) = SL(n, ℝ) are sharp up to finite loss of regularity. Moreover, we prove that for general \(\mathbb{G}\) the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that \(\mathbb{G}\) = SL(n, ℝ), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, ℝ) ⋉ ℝ2 and SL(2, ℝ) ⋉ ℝ4. A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

Asymptotic geometry in higher products of rank one Hadamard spaces

Given a product X of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups of X. First we give a detailed description of the structure of the geometric limit set and relate it to the limit cone; moreover, we show that the action of on a quotient of th e regular geometric boundary of X is minimal and proximal. This is completely analogous to the case of Zariski dense discrete subgroups of semi-simple Lie groups acting on the associated symmetric space (compare [Ben97]). In the second part of the paper we study the distribution of -orbit points in X: As a generalization of the critical exponent δ() of we consider for any θ ∈ R r0 , kθk = 1, the exponential growth rate δ�() of the number of orbit points in X with prescribed “slope” θ. In analogy to Quint’s result in [Qui02b] we show that the homogeneous extension � to R r0 of δ�() as a function of θ is upper semi-continuous, concave and strictly positive in the relative interior of the limit cone. This shows in particular that there exists a unique slope θ � for which δ��() is maximal and equal to the critical exponent of . We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ( [CS11, Theorem A]) for CAT(0)-cube complexes: Given a finite-dimensional CAT(0)-cube complex X and a group of automorphisms without fixed point in the geometric comp actification of X, then either contains a rank one isometry or there exists a conve x -invariant subcomplex of X which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex -invariant s ubcomplex of X which can be decomposed into a finite product of rank one Hadamard spaces. So our results imply in particular that classical properties of discrete subgroups of higher rank Lie groups as in [Ben97] and [Qui02b] also hold for certain discrete isometry groups of reducible CAT(0)-cube complexes.

A

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.

Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

Let \({\mathbb{G}}\) denote a higher-rank \({\mathbb{R}}\)-split simple Lie group of the following type: \({SL(n,\mathbb{R})}\), SOo(m, m), E6(6), E7(7) and E8(8), m ≥ 4. For any unitary representation \({(\pi,\mathcal{H})}\) of \({\mathbb{G}}\) without non-trivial \({\mathbb{G}}\)-invariant vectors, we study smooth solutions of the cohomological equation \({\mathfrak{u}f=g}\), where \({\mathfrak{u}}\) is a vector in the root space of \({\mathfrak{G}}\), the Lie algebra of \({\mathbb{G}}\) and g is a given vector in \({\mathcal{H}}\). We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for f. We also study common solutions to (the infinitesimal version of) the cocycle equation \({\mathfrak{u}h=\mathfrak{v}g}\), where \({\mathfrak{u}}\) and \({\mathfrak{v}}\) are commutative vectors in different root spaces of \({\mathfrak{G}}\) and g and h are given vectors in \({\mathcal{H}}\). We give the sufficient condition under which the cocycle equation has a common solution: (∗) if \({\mathfrak{u}}\) and \({\mathfrak{v}}\) embed in \({\mathfrak{sl}(2,\mathbb{R})\times \mathbb{R}}\), then the common solution exists. In fact, condition (∗) is also necessary when \({\mathbb{G}=SL(n,\mathbb{R})}\). We show counter examples in each \({SL(n,\mathbb{R})}\), n ≥ 3 if condition (∗) is not satisfied. Especially, the cocycle rigidity always fails for \({SL(3,\mathbb{R})}\). As an application, we obtain smooth cocycle rigidity for higher rank parabolic actions over \({\mathbb{G}/\Gamma}\) if the Lie algebra of the acting parabolic subgroup contains a pair \({\mathfrak{u}}\) and \({\mathfrak{v}}\) satisfying property (∗). The main new ingredient in the proof is making use of unitary duals of \({SL(2,\mathbb{R})\ltimes\mathbb{R}^2}\) and \({(SL(2,\mathbb{R})\ltimes\mathbb{R}^2)\ltimes\mathbb{R}^3}\) obtained by Mackey theory.

On Lie groups and hyperbolic symmetry—From Kunze–Stein phenomena to Riesz potentials

Sharp forms of Kunze–Stein phenomena on SL(2,R) are obtained by using symmetrization and Stein–Weiss potentials. A new structural proof with remarkable simplicity can be given on SL(2,R) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and n-dimensional hyperbolic space through application of the Riesz–Sobolev rearrangement inequality. A new framework is developed for Riesz potentials on semisimple symmetric spaces and the semi-direct product of groups analogous to the Iwasawa decomposition for semisimple Lie groups. Extensions to higher-rank Lie groups and analysis on multidimensional connected hyperboloids including anti de Sitter space are suggested by the analysis outlined here.

Asymptotic geometry in products of Hadamard spaces with rank one isometries

In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product $\XX=\XX_1\times \XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\delta_\theta(\Gamma)$ of orbit points in $\XX$ with a prescribed so-called "slope" $\theta\in (0,\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\Psi_\Gamma$ to $\RR_{\ge 0}^2$ of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous and concave.

Representations of surface groups and right-angled Artin groups in higher rank

We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and show that the criterion is satisfied in particular cases. There are direct applications towards constructing representations of surface groups into higher-rank Lie groups, and, in particular, into lattices in higher-rank Lie groups.

Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups

Let X be a globally symmetric space of noncompact type, \( G = \textrm{Isom}^o (X) \) and \( \Gamma \subset G \) a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary \( \theta X \) of \( \theta X \), we prove that for regular boundary points \( \xi \in \theta X \) , the Hausdorff dimension of the radial limit set in \( G \cdot \xi \) is bounded above by the exponential growth rate of the number of orbit points close in direction to \( G \cdot \xi \subseteq \theta X \). Furthermore, for Zariski dense discrete groups Γ we construct Γ-invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset \( G \cdot \xi \subseteq \theta X \).

Mostow-Margulis rigidity with locally compact targets

Let $ \Gamma $ be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which $ \Gamma $ (with the only exception of non-uniform lattices in $ G \simeq {\rm SL_2}({\Bbb R}) $ , which are virtually free groups) we give a similar description of all possible locally compact groups H, in which $ \Gamma $ can be embedded as a uniform lattice.

Rigidity, Unitary Representations of Semisimple Groups, and Fundamental Groups of Manifolds with Rank One Transformation Group

The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.

Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map H^2_b(\Gamma)\to H^2(\Gamma) from bounded to usual cohomology is injective if \Gamma is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for \Gamma : the stable commutator length vanishes and any C^1 -action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H^\bullet_b(\Gamma) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.

Spherical Functions of the Principal Series Representations of $Sp(2, \mathbf R)$ as Hypergeometric Functions of $\mathrm C_2$-Type

In this article we determine explicitly the systems of differential equations satisfied by spherical functions with non-trivial K-typQs of the principal series and the generalized principal series representations of Sp(2, R). Then we obtain series expansions and integral formulas of spherical functions of the generalized principal series representation. We shall define spherical functions. Let G be a real reductive Lie group and K be maximal compact subgroup of G, Po=MoA0No be a parabolic subgroup of G. Let Hn be an admissible representation of G and (r, Vr), (#, Vri) be irreducible representations of K which is contained in HK. We call elements of Horru( Vr, C7(K\G))^ C?(K\G)®KV? = C~,r(K\G/K) spherical functions of type-(#, r), where CTM(K\G) is the space of smooth sections of the homogeneous vector bundle over K\G associated to V? and V? is the contragredient representation of VT. Let 0E:Horri(ai K}(Hn, CTM(K\G}} and i^ Horrid Vr, HTC then °i is a spherical function attached to H*. There are many studies on the system of differential equations satisfied by spherical functions for 1-dimensional K-typzs. Moreover they are generalized as the Weyl group invariant commuting differential operators with continuous parameters, which are introduced by generalizing root multiplicities (cf. [DGl], [DG2], [HI], [HO], [Ko], [OO], [Opl], [Op2], [Os], [OS], [Sh]). On the other hand, there are few studies for vector-valued spherical functions. Besides, spherical functions are rarely calculated in explicit forms except for rank one cases. Therefore it is interesting to study vector-valued spherical functions of higher rank Lie group explicitly.

On systems of generators of arithmetic subgroups of higher rank groups

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of the congruence subgroup problem. We show that any two maximal opposing unipotent subgroups of an irreducible lattice in a higher rank Lie Group, generate a lattice in the Lie Group. The method of proof is to use certain techniques of the solution of the congruence subgroup problem of arithmetic lattices in higher rank groups. We freely use the notation and results of [3] without giving explicit references therein. Let G be a simply connected absolutely almost simple linear algebraic group defined and isotropic over a global field K. Let C/+ be the unimpotent radical (which is defined over K) of a minimal parabolic A'-subgroup P+ of G. Let U~~ be the unipotent radical of another minimal parabolic A'-subgroup P~ of G which is opposed to P+ in the sense that U+ Γ\U~ = {1}. Let S be a finite set of places of including all the archimedian ones, if any. We call thering A — Os — {x G K \x v < 1 for all places v of A', not in S} the ring of S-integers in K. Choose a faithful representation G <—» GLpj defined over and define G(Os) = {g G G gij G Os, 1 < i,j < N}. The subgroups in G which are of finite index in G(Os) are called S-arithmetic groups. Define the 5-rank of G to be the sum Kv — rank(G). Given a non-zero ideal α of A and an algebraic ves