Crystallographic classes on the minkowski space <Emphasis Type="Italic">R</Emphasis><Subscript>1,2</Subscript>. I. Theorems

Some preliminary statements are proved necessary for calculating the crystallographic groups in six crystallographic classes in the 3-dimensional Minkowski space. Three classes are determined by the unimodular subgroups of the general Lorentz group and three more classes, by subgroups unimodular in a certain isotropic coordinate system.     

Crystallographic classes in a 4-dimensional Minkowski space

On a crystallographic group, a condition of being topologically discrete is imposed which is weaker than is the conventional requirement for an action on space to be discontinuous. Isomorphism classification is given for crystallographic groups in three crystallographic classes in a 4-dimensional Minkowski space, which are defined by unimodular subgroups of the general Lorentz group. In these classes are, respectively, 24, 36, and 68 crystallographic groups. __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 31–53, January–February, 2008.     

A method for obtaining representatives of classes of Maxwell spaces

Four classes of potential structures on Minkowski space which admit subgroups of the Poincaré group are constructed. These classes are used to obtain representatives of classes of Maxwell spaces with the same symmetry groups. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Quasicrystallographic groups on Minkowski spaces

We generalize the quasicrystallographic groups in the sense of Novikov and Veselov from Euclidean spaces to pseudo-Euclidean and affine spaces. We prove that the quasicrystallographic groups on Minkowski spaces whose rotation groups satisfy an additional assumption are projections of crystallographic groups on pseudo-Euclidean spaces. An example shows that the assumption cannot be dropped. We prove that each quasicrystallographic group is a projection of a crystallographic group on an affine space.     

The structure of orthogonal groups of 2-adic unimodular quadratic forms

This paper gives some indication of the structure of the orthogonal group of a unimodular quadratic form over the 2-adic integers. Various congruence subgroups are defined and their interrelationships investigated. Figure 1 gives the lattice pattern of the subgroups considered when the rank is even. The subgroups normalized by the commutator subgroup are also determined.     

Variations on the theme of FC-groups

The following is clearly equivalent to the usual definition of FC-group. A group is an FC-group, if each of its cyclic subgroups has only finitely many conjugates. We consider several weaker conditions on the conjugates of cyclic subgroups, the strongest of which we show is equivalent to the FC-condition for many classes of groups.     

Measurable groups of low dimension

We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any 2‐dimensional asymptotic group is soluble‐by‐finite. We obtain a field‐interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.     

The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces

The weak Bieberbach theorem states that each crystallographic group on a Euclidean space uniquely determines its translation lattice as an abstract group. Garipov proved in 2003 that the same holds for crystallographic groups on Minkowski spaces and asked whether a similar claim holds in the pseudo-Euclidean spaces ℝ ^p,q . We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces ℝ ^p,q with min{p, q} ≤ 2. For min{p, q} ≥ 3 we construct examples of crystallographic groups with two distinct lattices exchanged by a suitable automorphism of the group. For crystallographic groups with two distinct isomorphic pseudo-Euclidean lattices we also prove that the coranks of their intersection in these lattices can take arbitrary values greater than 2 with the exception of 4.     

Pseudo‐c‐archimedean and pseudo‐finite cyclically ordered groups

Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.     

Groups with few conjugacy classes of non-normal subgroups

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.      

On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.     

非幂零真子群同阶类类数给定的有限群

作为Schmidt定理的推广,证明了:(1)非幂零真子群同阶类类数<3的有限群可解;(2)G为非幂零真子群同阶类类数=3的非可解群当且仅当G≌A_5或G≌SL_2(5).此外,完全分类了非平凡幂零子群同阶类类数≤5的非可解群和非平凡子群同阶类类数≤9的非可解群.     

A note on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties

Monica Klein classified Minkowski planes with respect to linearly transitive subgroups of Minkowski homotheties. She obtained 23 possible types. In this paper we investigate Minkowski planes with respect to groups of automorphism of certain Klein types 12 and higher. We show that types 12 and 14 can only occur in finite miquelian Minkowski planes of order 3 or 5, and we provide examples for such groups. Furthermore, we prove that types 13 and 18 in finite Minkowski planes can only occur in miquelian planes.     

矩阵环的有限乘法子群

将Minkowski关于有限整数矩阵群的著名结果推广到一般的环上,主要结果是证明了:对任意环R,如果R的加法群为有限生成的自由Abel群,则R的所有乘法可逆元构成的群U(R)中的有限子群精确到同构只有有限多个.     

Enumeration of maps regardless of genus: Geometric approach

We use the conceptual idea of “maps on orbifolds” and the theory of the non-Euclidean crystallographic groups (NEC groups) to enumerate rooted and unrooted maps (both sensed and unsensed) on surfaces regardless of genus. As a consequence we deduce a formula for the number of chiral pairs of maps. The enumeration principle used in this paper is due to Mednykh (2006) [15], it counts the number of conjugacy classes of subgroups in NEC groups which are in one-to-one correspondence with unrooted (sensed or unsensed) maps.     

Ornament Groups on a Minkowski Plane

We are engaged in classifying up to isomorphism of discrete subgroups of an affine transformation group on a plane (a two-dimensional space) preserving the Minkowski metric. It is proved that, for subgroups that do not coincide with Euclidean ones, the orbit of almost every point is everywhere dense.     

Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.