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.     

Rectifiability and Lipschitz extensions into the Heisenberg group

Denote by \mathbbHⁿ{\mathbb{H}^n} the 2n + 1 dimensional Heisenberg group. We show that the pairs (\mathbbR^k ,\mathbbHⁿ){(\mathbb{R}^k ,\mathbb{H}^n)} and (\mathbbH^k ,\mathbbHⁿ){(\mathbb{H}^k ,\mathbb{H}^n)} do not have the Lipschitz extension property for k  >  n.     

Renormalization group in turbulence theory: Exactly solvable Heisenberg model

An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε=2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 245–262 May. 1998.     

On weakly-central group extensions

Some properties of weakly-central extensions for which the quotient group by the kernel is hypercentral are investigated. The existence of a hypercentral coradical in any such periodic group is proved and the complementability of some of its subgroups in the group is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1017–1021, July–August, 1991.     

On the IYB-property in some solvable groups

A finite group G is called Involutive Yang-Baxter (IYB) if there exists a bijective 1-cocycle ${\chi: G \longrightarrow M}$ for some ${\mathbb{Z}G}$ -module M. It is known that every IYB-group is solvable, but it is still an open question whether the converse holds. A characterization of the IYB property by the existence of an ideal I in the augmentation ideal ${\omega\mathbb{Z}G}$ complementing the set 1?G leads to some speculation that there might be a connection with the isomorphism problem for ${\mathbb{Z}G}$ . In this paper we show that if N is a nilpotent group of class two and H is an IYB-group of order coprime to that of N, then ${N \rtimes H}$ is IYB. The class of groups that can be obtained in that way (and hence are IYB) contains in particular Hertweck’s famous counterexample to the isomorphism conjecture as well as all of its subgroups. We then investigate what an IYB structure on Hertweck’s counterexample looks like concretely.     

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.     

On extensions of some block designs

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.