Nilpotent Lie Groups Research Articles

Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

Let $G$ be a connected, simply connected one-parameter metabelian nilpotent Lie group, that means, the corresponding Lie algebra has a one-codimensional abelian subalgebra. In this article we show that $G$ contains a discrete cocompact subgroup. Given a discrete cocompact subgroup $\Gamma$ of $G$, we define the quasi-regular representation $\tau = {\rm ind}_\Gamma^G 1$ of $G$. The basic problem considered in this paper concerns the decomposition of $\tau$ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between $\tau$ and its desintegration without considering multiplicities. Finally, unlike the Moore inductive algorithm for multiplicities on nilmanifolds, we carry out here a direct computation to get the multiplicity formula.

An Upper Bound for the Poisson Kernel on Higher Rank NA Groups

Let S be a semi direct product \(S=N\rtimes A\) where N is a connected and simply connected nilpotent Lie group and A is isomorphic with ℝk, k > 1. We obtain an upper bound for the Poisson kernel for the class of second order left-invariant differential operators on S.

Continuity of magnetic Weyl calculus

We investigate continuity properties of the operators obtained by the magnetic Weyl calculus on nilpotent Lie groups, using modulation spaces associated with unitary representations of certain infinite-dimensional Lie groups.

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.

On topological conjugacy of left invariant flows on semisimple and affine Lie groups

In this paper, we study the flows of nonzero left invariant vector fields on Lie groups with respect to topological conjugacy. Using the fundamental domain method, we are able to show that on a simply connected nilpotent Lie group any such flows are topologically conjugate. Combining this result with the Iwasawa decomposition, we find that on a noncompact semisimple Lie group the flows of two nilpotent or abelian fields are topologically conjugate. Finally, for affine groups G = HV, V ≅ n, we show that the conjugacy class of a left invariant vector field does not depend on its Euclidean component.

The Ricci flow for simply connected nilmanifolds

We prove that the Ricci flow g(t) starting at any metric on R that is invariant by a transitive nilpotent Lie group N can be obtained by solving an ODE for a curve of nilpotent Lie brackets on R. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))| g(t) converges to a Ricci soliton in C∞, uniformly on compact sets in R. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to N .

A smooth family of intertwining operators

Let $N$ be a connected, simply connected nilpotent Lie group with Lie algebra $\mathfrak{n}$ and let $\mathscr{W}$ be a submanifold of $\mathfrak{n}^*$ such that the dimension of all polarizations associated to elements of $\mathscr{W}$ is fixed. We choose $(\mathfrak{p}(w))_{w \in \mathscr{W}}$ and $(\mathfrak{p}'(w))_{w \in \mathscr{W}}$ two smooth families of polarizations in $\mathfrak{n}$. Let $\pi_w = \mathsf{ind}_{P(w)}^N \chi_w$ and $\pi'_w = \mathsf{ind}_{P'(w)}^N \chi_w$ be the corresponding induced representations, which are unitary and irreducible. It is well known that $\pi_w$ and $\pi'_w$ are unitary equivalent. In this paper, we prove the existence of a smooth family of intertwining operator $(T_w)_w$ for theses representations, where $w$ runs through an appropriate non-empty relatively open subset of $\mathscr{W}$. The intertwining operators are given by an explicit formula.

Nilpotent groups without exactly polynomial Dehn function

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of nilpotent Lie groups. This class contains groups for which it is known that their Dehn function grows no faster than n2log n. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

On the classification of weakly symmetric Finsler spaces

In this paper, we give the classification of some special types of weakly symmetric Finsler spaces. We first present a general principle to classify weakly symmetric Finsler spaces and also give a method to figure out the Berwald spaces among the class of weakly symmetric Finsler spaces. Then we give an explicit classification of weakly symmetric Finsler spaces with reductive isometric groups as well as the left invariant weakly symmetric Finsler metrics on nilpotent Lie groups of the Heisenberg type. As an application, we obtain a large number of high-dimensional examples of reversible Finsler spaces which are non-Berwaldian and with vanishing S-curvature, a kind of spaces which are sought after in an open problem of Z. Shen.

Maximal holonomy of infra-nilmanifolds with 3-dimensional Iwasawa geometry

Let be the nilpotent Lie group of all unipotent upper triangular complex n × n matrices and let In denote the maximal order of the holonomy groups of all infranilmanifolds with -geometry. By analyzing the algebraic structure of and studying its isometry group, we prove that I3 = 24.

Cowling Price Theorem for Low Dimensional Nilpotent Lie Groups

We extend an uncertainty principle due to Cowling and Price to low dimensional Nilpotent Lie groups G4. The uncertainty principle is age neralization of a classical result due to Hardy.

An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups

This note extends the classical theorem of Minkowski on lattice points and convex bodies in ℝ^n to 2-step simply connected nilpotent Lie groups with a ℚ -structure. This includes all groups of Heisenberg type. More generally (and more naturally), it works for any simply connected nilpotent Lie group with a ℚ -structure whose Lie algebra admits a grading of length 2. Here a new invariant associated with the grading occurs which we call the degree . It explains why some directions are more equal than others.

Estimate of the Lp -Fourier transform norm for connected nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group, and m the dimension of the generic coadjoint orbits of G . Then it is proved in [Baklouti, Smaoui, Ludwig, J. Funct. Anal. 199: 508–520, 2003] that the operator value Fourier transform norm satisfies , where and q being the conjugate of p . When the assumption on G to be simply connected is removed, we prove an analogue estimate of type , where H designates the maximal compact central subgroup of G and m the dimension of generic coadjoint orbits of the universal covering of G . The case of non-simply connected Heisenberg groups is treated as an example.

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.

Remarks on a conjecture of Chabauty

The Chabauty conjecture for connected nilpotent Lie groups has been proved by S. P. Wang. We show that one reasoning flaw has infiltrated the proof. We therefore give a new proof of the validity of Chabauty's conjecture in this setup. More generally, we shall prove that the Chabauty conjecture is true for rigid lattices and Zariski dense lattices of connected solvable Lie groups. In particular, the Chabauty conjecture holds for solvable Lie groups of (R)-type.

Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups

Assume that $X$ is a hyperbolic basic set for $f:X\to X$.We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensionsof $f$, those that are transitive are open and dense. The fibers are semidirectproducts of compact and nilpotent groups.

GEODESIC FORMULA OF A CERTAIN CLASS OF PSEUDORIEMANNIAN 2-STEP NILPOTENT GROUPS AND JACOBI OPERATORS ALONG GEODESICS IN PSEUDORIEMANNIAN 2-STEP NILPOTENT GROUPS

In this paper, we obtain geodesic formula of a certain class of Pseudoriemmanian 2-step nilpotent groups and show a constancy of represenation matrix of Jacobi oprerators along geodesics in Pseudoriem- manian 2-step nilpotent groups with one dimensional center. These are nonabelian Lie groups that come as close as possible to being abelian, but they admit interesting phenomena that do not arise in abelian groups. During the 1990's many works were done in 2-step nilpotent Lie groups with a left invariant Riemmannian metric or Lorentzian metric. Eberlein (1) and Guederi (2) did general studies on 2-step nilpotent Lie groups with a left invariant Riemman- nian metric and 2-step nilpotent Lie groups with a left invariant Lorentzian metric, respectively. But relatively little was known of the Pseudoriemmanian case. For example , Eberlein and Guederi succeeded in obtaining general ge- odesic formulae of 2-step nilpotent groups with a left invariant Riemmanian metric and with a left invariant Lorentzian metric respectively. But until now a geodesic formula of Pseudoriemmanian 2-step nilpotent groups has not been obtained. In this paper, we obtain a geodesic formula of a special famliy of Pseudoriemmanian 2-step nilpotent groups and show a constancy of represen- ation matrix of Jacobi oprerators along geodesics in Pseudoriemmanian 2-step nilpotent groups with one dimensional center, which is a slight generalization of a result in (6).

Localized Hardy spaces 𝐻¹ related to admissible functions on RD-spaces and applications to Schrödinger operators

Let X {\mathcal X} be an RD-space, which means that X {\mathcal X} is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X {\mathcal X} . In this paper, the authors first introduce the notion of admissible functions ρ \rho and then develop a theory of localized Hardy spaces H ρ 1 ( X ) H^1_\rho ({\mathcal X}) associated with ρ \rho , which includes several maximal function characterizations of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) , the relations between H ρ 1 ( X ) H^1_\rho ({\mathcal X}) and the classical Hardy space H 1 ( X ) H^1({\mathcal X}) via constructing a kernel function related to ρ \rho , the atomic decomposition characterization of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) , and the boundedness of certain localized singular integrals on H ρ 1 ( X ) H^1_\rho ({\mathcal X}) via a finite atomic decomposition characterization of some dense subspace of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) . This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R n \mathbb {R}^n , or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.

The geometry of filiform nilpotent Lie groups

We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We describe the connection and curvatures, and we investigate necessary and sufficient conditions for subgroups to be totally geodesic submanifolds. We also classify the one-parameter subgroups which are geodesics. Department of Mathematics, Wellesley College, 106 Central St., Wellesley, MA 02481-8203 mkerr@wellesley.edu Department of Mathematics, Idaho State University, Pocatello, ID 83209-8085 payntrac@isu.edu, tpayne@member.ams.org

Foucault pendulum and sub-Riemannian geometry

The well known Foucault nonsymmetrical pendulum is studied as a problem of sub-Riemannian geometry on nilpotent Lie groups. It is shown that in a rotating frame a sub-Riemannian structure can be naturally introduced. For small oscillations, three dimensional horizontal trajectories are computed and displayed in detail. The fiber bundle structure is explicitly shown. The underlying Lie structure is described together with the corresponding holonomy group, which turns out to be given by the center of the Heisenberg group. Other related physical problems that can be treated in a similar way are also mentioned.