Lie Group Of Real Rank Research Articles

On the dimension drop conjecture for diagonal flows on the space of lattices

On the dimension drop conjecture for diagonal flows on the space of lattices

Linearity and Indiscreteness of Amalgamated Products of Hyperbolic Groups

Abstract We discuss the linearity and discreteness of amalgamated products of linear word hyperbolic groups. In particular, we prove that the double of a torsion-free Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups, which do not admit discrete and faithful representations into any simple Lie group of real rank $1$. We also build new examples of non-linear word hyperbolic groups, elaborating on a previous work of Canary–Stover–Tsouvalas.

Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers

Let X be a rank one Riemannian symmetric space of noncompact type and Δ be the Laplace–Beltrami operator of X . The space X can be identified with the quotient space G / K where G is a connected noncompact semisimple Lie group of real rank one with finite center and K is a maximal compact subgroup of G . Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK ), when it is invariant under the left-action of K . We consider right-convolution operators Θ on functions f on X defined by, Θ : f ↦ f ⁎ μ where μ is a radial (possibly complex) measure on X . These operators will be called multipliers. In particular Θ is a radial average when μ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages and the heat operator. In this paper we address problems of the following type: Fix a multiplier, in particular an averaging operator Θ. Suppose that { f k } k ∈ Z is a bi-infinite sequence of functions on X such that for all k ∈ Z , Θ f k = A f k + 1 and ‖ f k ‖ < M for some constants A ∈ C , M > 0 and a suitable norm ‖ ⋅ ‖ . From this hypothesis, we try to infer that f 0 , hence every f k , is an eigenfunction of Δ.

Mini-Workshop: Reflection Groups in Negative Curvature

Discrete groups generated by reflections constitute an important source of examples of lattices in simple Lie groups of real rank 1 (whose associated symmetric spaces are negatively curved). Yet a classification for them is far from being achieved, even in the case of hyperbolic geometry. The goal of this mini-workshop was to stimulate the research by bringing together specialists of different aspects of the theory.

Arithmeticity of discrete subgroups containing horospherical lattices

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.

Lattices in 𝑃𝑈(𝑛,1) that are not profinitely rigid

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in PU ⁡ ( n , 1 ) \operatorname {PU}(n,1) with isomorphic profinite completions for all n ≥ 2 n \ge 2 . This disproves a conjecture of D. Kazhdan and gives the first examples of nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.

A shrinking target problem with target at infinity in rank one homogeneous spaces

In this paper, we give a definition of Diophantine points of type $\gamma$ for $\gamma\geq0$ in a homogeneous space $G/\Gamma$, and compute the Hausdorff dimension of the subset of points which are not Diophantine of type $\gamma$ when $G$ is a semisimple Lie group of real rank one. We also deduce a Jarn\'ik-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.

Counting non-uniform lattices

In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x(γ(H)+o(1)) log x/ log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In [BLu] we disproved this conjecture. In this paper we prove that for most groups H the conjecture is actually true if we restrict to counting only non-uniform lattices.

How to prove the Baum–Connes conjecture for the groups [formula omitted] ?

Abstract We present a programme of proof of the Baum–Connes conjecture with coefficients for G=Sp(n,1) or F4(-20), i.e. simple Lie groups of real rank one having Kazhdan’s property (T). We use the geometry of the boundary sphere to produce a G-Fredholm module, together with a homotopy to the trivial representation through uniformly bounded representations. The strip of uniformly bounded representations of M. Cowling plays here the role of the complementary series. We explain how, modulo some conjectural estimates, this construction would prove the conjecture.

Rigidity for Von Neumann Algebras Given by Locally Compact Groups and Their Crossed Products

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.

Hyperbolic geometry and pointwise ergodic theorems

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

Fourier algebras of parabolic subgroups

We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.

Counting Lattice Points in norm balls on higher rank simple Lie groups

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.

Absence of two Paley–Wiener properties for semisimple Lie groups of real rank one

The weak Paley–Wiener property and the topological Paley–Wiener property for connected semisimple Lie groups of real rank one with finite center are discussed.

Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.

Escape of mass and entropy for diagonal flows in real rank one situations

Let G be a connected semisimple Lie group of real rank 1 with finite center, let Γ be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space Γ\G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full.

Helgason-Schiman Formula for Semisimple Lie Groups of Arbitrary Rank

This paper extends the Helgason-Schiffman formula for the H-function on a semisimple Lie group of real rank one to cover a semisimple Lie group G of arbitrary real rank. A set of analytic R-valued cocycles are reduced for certain real rank one subgroups of G. This allows a formula for the c-function on G to be worked out as an integral of a product of their resolutions on the summands in a direct-sum decomposition of the maximal abelian subspace of the Lie algebra g of G. Results about the principal series of representations of the real rank one subgroups are also obtained, among other things.

Manifolds counting and class field towers

In Burger et al. (2002) [12] and Goldfeld et al. (2004) [17] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x(γ(H)+o(1))logx/loglogx where γ(H) is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate xclogx. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.

Actions of higher-rank lattices on free groups

AbstractIf G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation ΔNu + (1 + ɛK(x, z))u2*−1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn’s subelliptic context on N in presence of symmetry.