Free Nilpotent Lie Group Research Articles

Anosov diffeomorphisms on infra-nilmanifolds modeled on a free 2-step nilpotent Lie group

In this article we study the existence question of Anosov diffeomorphisms on an infra-nilmanifold. After establishing a general existence criterion in terms of the associated holonomy representation, we concentrate on infra-nilmanifolds for which the covering Lie group is a free nilpotent Lie group. In turns out that in this case the criterion obtained before can be reduced drastically. Finally, we completely solve the existence question in case the covering Lie group is free 2-step nilpotent and the holonomy group is abelian.

Anosov diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group

In this article we give an elementary proof, using standard arguments from algebraic number theory, of the fact that a nilmanifold modelled on a free c-step nilpotent Lie group on n generators admits an Anosov diffeomorphism if and only if n > c. In fact, we need to show that for any integer n > 1, there exists a matrix A ∈ GL(n, ℤ), such that any product of less than n eigenvalues of A is of modulus ≠ 1.

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set \( {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) of K-orbits in the dual \( \mathfrak{n}* \) of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence \( \Delta {\left( {K,N} \right)} \leftrightarrow {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.

On the best constant for the Friedrichs-Knapp-Stein inequality in free nilpotent Lie groups of step two and applications to subelliptic PDE

In this article we propose to find the best constant for the Friedrichs-Knapp-Stein inequality in F2n,2, that is the free nilpotent Lie group of step two on 2n generators, and to prove the second-order differentiability of subelliptic p-harmonic functions in an interval of p.

ANOSOV DIFFEOMORPHISMS ON A CLASS OF 2-STEP NILMANIFOLDS

In this paper we study nilmanifolds which are modeled on a quotient of a free 2-step nilpotent Lie group by a 1-dimensional subgroup. In fact we obtain a very easy criterion to decide whether or not such a nilmanifold admits an Anosov diffeomorphism.

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations

In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family Lξ 0 of left invariant operators on a free nilpotent Lie group. The fundamental solution Γξ 0 of the operator Lξ 0 is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is C∞.

Links, pictures and the homology of nilpotent groups

We give a geometric slice-like characterization for the vanishing of Milnor's link invariants by proving the k-slice conjecture. This conjecture states that a link L has vanishing Milnor μ ̄ -invariants of length ⩽2 k if and only if L bounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent quotients of class k. In the course of our proof, we compute the dimension ⩽3 homology groups of finitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighted chain resolution for a nilpotent group with torsion free lower central series quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new derivation of the May spectral sequence relating the group homology of the nilpotent group to the Lie ring homology of its associated graded Lie ring. Finally, we define μ ̄ -invariants of “pictures” and use these to describe a generating set of cocycles in the cohomology of the free nilpotent groups. Some sample computations follow.

Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in G L ( n , Z ) \mathrm {GL}(n,\mathbb {Z}) whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of Q K \mathbb {Q}^K , a subject which is of interest on its own.

Quadratic presentations and nilpotent Kähler groups

It has been known for at least thirty years that certain nilpotent groups cannot be Kahler groups, i.e., fundamental groups of compact Kahler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-step nilpotent Lie groups, k > 1, are not Kahler. The Heisenberg case was known to J-P. Serre in the early 1960’s, and unified proofs of the above statements follow readily from Sullivan’s theory of minimal models [6],[15], [19], or from Chen’s theory of iterated integrals [4], [10], or from more recent developments such as [9].

Remarks on manifolds which admit locally free nilpotent Lie group actions

Remarks on manifolds which admit locally free nilpotent Lie group actions