Homogeneous geodesics in solvable Lie groups

After the second author and J. Szenthe [10] proved that every homogeneous Riemannian manifold admits a homogeneous geodesic, several authors studied the set of all homogeneous geodesics in various homogeneous spaces. In this paper, we consider special examples of homogeneous spaces of solvable type of arbitrary odd dimension given in [1] and [7] and we show that their sets of homogeneous geodesics have an interesting structure, closely connected to the notion of Hadamard matrices.     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Isoperimetry of nilpotent groups

This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.     

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests.     

Topological properties of the multiplication in a nilpotent Lie group

LetG be a simply connected nilpotent Lie group. IfK andH are closed connected subgroups which intersect only in the neutral element, the multiplicationK ×H G is shown to be a proper mapping. Furthermore, we consider the operation ofK ×H onG given byg · (k, h)=k ^–1 gh. It is shown that, under certain assumptions, this operation is proper if and only if it is free. Under more restrictive assumptions, the quotientKG/H is diffeomorphic to an ⁿ .     

Rigidity problem for lattices in solvable Lie groups

The paper concerns rigidity problem for lattices in simply connected solvable Lie groups. A lattice Γ⊂G is said to be rigid if for any isomorphism ϕ:Γ→Γ′ with another lattice Γ′⊂G there exists an automorphism which extends ϕ. An effective rigidity criterion is proved which generalizes well-known rigidity theorems due to Malcev and Saito. New examples of rigid and nonrigid lattices are constructed. In particular, we construct: a) rigid lattice Γ⊂G which is not Zariski dense in the adjoint representation ofG, b) Zariski dense lattice Γ⊂G which is not rigid, c) rigid virtually nilpotent lattice Γ in a solvable nonnilpotent Lie groupG.     

The accuracy of Gaussian approximations in nilpotent Lie groups

We consider the Central Limit Theorem with Gaussian limit distributions in stratified nilpotent Lie groups. We obtain estimates of the rate of convergence and Edgeworth expansions for expectations of smooth functionals.     

The behavior of Fourier transforms for nilpotent Lie groups

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on extends to be ``holomorphic' on an appropriate complexification of (a part of) . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of when is two-step nilpotent.

     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Hardy spaces on Lie groups of polynomial growth

We give several characterizations of Hardy spaces associated with complex, second-order,subelliptic operators on Lie groups with polynomial growth.     

Computing the test rank of a free solvable Lie algebra

We computed the test rank of a free solvable Lie algebra of finite rank.     

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g₀ we associate a Clifford algebra over the field of rational functions on O. We find the rank, k(O) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U(g)-module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k(O) is in many cases, equal to the odd dimension of the orbit G⋅O, where G is a Lie supergroup with Lie superalgebra g.     

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.     

Completely simple semigroups with nilpotent structure groups

In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing [x ₁,x ₂,…,x _c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case. The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Averaged Dehn functions for nilpotent groups

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality δ(l)<Cl^α for α>2, then it satisfies the averaged isoperimetric inequality . In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.     

On the second cohomology of nilpotent orbits in exceptional Lie algebras

In [1], the second de Rham cohomology groups of nilpotent orbits in all the complex simple Lie algebras are described. In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.