Rectifiability and perimeter in the Heisenberg group

In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Carathéodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems. Received: 27 March 2000 / Revised version: 13 December 2000 / Published online: 24 September 2001     

Lipschitz continuity of refinable functions on the Heisenberg group

In this paper, the Lipschitz continuity of refinable functions related to the general acceptable dilations on the Heisenberg group will be investigated in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces related to a kind of special acceptable dilations.     

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the $L^{\infty }$ -norm of its intrinsic gradient.     

Homogeneous Riemannian structures on some solvable extensions of the Heisenberg group

Two families of four or five-dimensional Riemannian solvable Lie groups, which are extensions of the Heisenberg group, are considered. We determine all the homogeneous Riemannian structures on them, and the simply connected groups of isometries corresponding to the associated reductive decompositions. Some of these structures are homogeneous Kähler or homogeneous cosymplectic, and in these cases they are realized by the complex hyperbolic plane ?H(2) and by ?H(2)×?, respectively.     

Optimal Lipschitz extensions and the infinity laplacian

We reconsider in this paper boundary value problems for “infinity Laplacian” PDE and the relationships with optimal Lipschitz extensions of the boundary data. fairly elegant new proofs, which clarify and simplify previous work, and may be characterized by a comparison principle with appropriate cones. We in comparison with cones directly implies the variational principle associated Liouville theorem for subsolutions bounded above by planes. Received: 29 May 2000 / Accepted: 12 June 2000 / Published online: 23 April 2001     

Assouad-Nagata dimension via Lipschitz extensions

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov and capacity dimension of Buyalo [7]) to Assouad-Nagata dimension. This is done by applying two functors on the Lipschitz category of metric spaces: microscopic and macroscopic. In the second part we identify (among spaces of finite Assouad-Nagata dimension) spaces of Assouad-Nagata dimension at most n as those for which the n-sphere S ⁿ is a Lipschitz extensor. Large scale and small scale analogues of that result are given. The author was partially supported by Grant No.2004047 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The author was supported by Grant AP2004-2494 from the Ministerio de Educacion y Ciencia, Spain. He thanks the Department of Mathematics of University of Tennessee for their hospitality.     

Lipschitz Homotopy Groups of the Heisenberg Groups

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ${\mathbb{H}^n}$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj?asz-Lukyanenko-Tyson constructed horizontal maps from S ^k to ${\mathbb{H}^n}$ which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map ${S^k \to \mathbb{H}^1}$ factors through a tree and is thus Lipschitz null-homotopic if ${k \geq 2}$ .     

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.     

Compensated compactness and the Heisenberg group

Work partially supported by the National Science Foundation     

Quasiconvexity in the Heisenberg group

We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.     

Box spaces,group extensions and coarse embeddings into Hilbert space

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.     

Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

On the torsion of group extensions and group actions

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z_p^r may act freely on (S¹)^k if and only if r?k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689-2700] for central extensions.