Engel Group Research Articles

Spectral summability for the quartic oscillator with applications to the Engel group

In this article, we investigate spectral properties of the sublaplacian -\Delta_{G} on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying F(-\Delta_{G})u=u\star k_{F} , for suitable scalar functions F , and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.

Unextendable Intrinsic Lipschitz Curves:

In the setting of Carnot groups, we exhibit examples of intrinisc Lipschitz curves of positive $\mathcal{H}^1$-measure that intersect every connected intrinsic Lipschitz curve in a $\mathcal{H}^1$-negligible set. As a consequence such curves cannot be extended to connected intrinsic Lipschitz curves. The examples are constructed in the Engel group and in the free Carnot group of step 3 and rank 2. While the failure of the Lipschitz extension property was already known for some pairs of Carnot groups, ours is the first example of the analogous phenomenon for intrinsic Lipschitz graphs. This is in sharp contrast with the Euclidean case.

An alternative construction of the Rumin complex on homogeneous nilpotent Lie groups

In this paper, we consider the Rumin complex on homogenenous nilpotent Lie groups. We present an alternative construction to the classical one on Carnot groups using ideas from parabolic geometry. We also give the explicit computations for the Engel group with this approach.

Mollifications of contact mappings of Engel group

The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as H\"older continuity, differentiability almost everywhere in the sense of Pansu, Luzin $\mathcal{N}$-property.

Cut time in the sub-Riemannian problem on the Cartan group

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.

Optimal horizontal joinability on the Engel group

On the Engel group we solve the problem of finding the minimal number of segments of integral lines of left-invariant basis horizontal vector fields necessary for joining an arbitrary pair of points. We prove the best version of Rashevskii–Chow Theorem on Engel group.

Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.

Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem

We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.

Extremals of a Left-Invariant Sub-Finsler Metric on the Engel Group

Using the Pontryagin maximum principle for the time-optimal problem in coordinates of the first kind, we find the extremals of an arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank 2.

GROUP ALGEBRAS WHOSE UNIT GROUP IS LOCALLY NILPOTENT

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

Hilbert-Haar coordinates and Miranda's theorem in Lie groups

We study the interior regularity of solutions to a class of quasilinear equations of non-degenerate p-Laplacian type on Lie groups that admit a system of Hilbert-Haar coordinates. These are coordinates with respect to which every linear function has zero symmetrized second order horizontal derivatives. All Carnot groups of rank two are in this category, as well as the Engel group, the Goursat type groups, and those general Carnot groups of step three for which the non-zero commutators of order three are linearly independent.

On (𝑛 + ½)-Engel groups

Abstract Let n be a positive integer. We say that a group G is an ( n + 1 2 ) {(n+\frac{1}{2})} -Engel group if it satisfies the law [ x , y n , x ] = 1 {[x,{}_{n}y,x]=1} . The variety of ( n + 1 2 ) {(n+\frac{1}{2})} -Engel groups lies between the varieties of n-Engel groups and ( n + 1 ) {(n+1)} -Engel groups. In this paper, we study these groups, and in particular, we prove that all ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel { 2 , 3 } {\{2,3\}} -groups are locally nilpotent. We also show that if G is a ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel p-group, where p ≥ 5 {p\geq 5} is a prime, then G p {G^{p}} is locally nilpotent.

Geodesics in the Engel Group with a Sub-Lorentzian Metric — the Space-Like Case

Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.

Groups which satisfy a Thue–Morse identity

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity [Formula: see text]. We prove that if [Formula: see text] is a 3-Thue–Morse group then [Formula: see text] is soluble for every [Formula: see text] and [Formula: see text] in [Formula: see text]. Furthermore, if [Formula: see text] is an Engel group without involution then we show that [Formula: see text] is locally nilpotent.

Sub-Finsler structures on the Engel group

A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.

Engel groups with an identity

We give an affirmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group [Formula: see text] with an identity, we prove that the set of right Engel elements of [Formula: see text] is contained in the Hirsch–Plotkin radical of [Formula: see text]. Given an arbitrary word [Formula: see text], we also show that the class of all groups [Formula: see text] in which the [Formula: see text]-values are right [Formula: see text]-Engel and [Formula: see text] is locally nilpotent is a variety.

Extremal trajectories in the sub-Lorentzian problem on the Engel group

Let be the Engel group and let be a rank-two left-invariant distribution with Lorentzian metric on . The sub-Lorentzian problem is stated as the problem of maximizing the sub-Lorentzian distance. A parametrization of timelike and spacelike normal extremal trajectories is obtained in terms of Jacobi elliptic functions. Discrete symmetry groups are described in the cases of timelike and spacelike trajectories; in both cases the fixed points and the corresponding Maxwell points are calculated for each symmetry. These calculations underlie estimates for the cut time (when the trajectory ceases to be globally optimal). Bibliography: 17 titles.

Control Systems on the Engel Group

We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.

On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation

It is shown that over an arbitrary field there exists a nil algebra R R whose adjoint group R o R^{o} is not an Engel group. This answers a question by Amberg and Sysak from 1997. The case of an uncountable field also answers a recent question by Zelmanov. In 2007, Rump introduced braces and radical chains A n + 1 = A ⋅ A n A^{n+1}=A\cdot A^{n} and A ( n + 1 ) = A ( n ) ⋅ A A^{(n+1)}=A^{(n)}\cdot A of a brace A A . We show that the adjoint group A o A^{o} of a finite right brace is a nilpotent group if and only if A ( n ) = 0 A^{(n)}=0 for some n n . We also show that the adjoint group A o A^{o} of a finite left brace A A is a nilpotent group if and only if A n = 0 A^{n}=0 for some n n . Moreover, if A A is a finite brace whose adjoint group A o A^{o} is nilpotent, then A A is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that A o A^{o} is sometimes called the multiplicative group of a brace A A . We also introduce a chain of ideals A [ n ] A^{[n]} of a left brace A A and then use it to investigate braces which satisfy A n = 0 A^{n}=0 and A ( m ) = 0 A^{(m)}=0 for some m , n m, n . We also describe connections between our results and braided groups and the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. It is worth noticing that by a result of Gateva-Ivanova braces are in one-to-one correspondence with braided groups with involutive braiding operators.

Cut Locus in the Sub-Riemannian Problem on Engel Group

This paper considers the left-invariant sub-Riemannian problem on the Engel group. The stratification of the cut locus and the structure of optimal synthesis are described.