Algebraic sets and coordinate groups for a free nilpotent group of nilpotency class 2

We give a complete classification of the algebraic sets and coordinate groups for the systems of equations in one variable over a free nilpotent group.     

Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Supported by RFBR (project No. 09-01-00099). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 309–341, May–June, 2009.     

The twisted conjugacy problem for endomorphisms of metabelian groups

Let M be a finitely generated metabelian group explicitly presented in a variety of all metabelian groups. An algorithm is constructed which, for every endomorphism φ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M′ and for any pair of elements u, v ∈ M, decides if an equation of the form (xφ)u = vx has a solution in M. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group M is decidable for an arbitrary endomorphism φ ∈ End(M). Supported by RFBR (project No. 07-01-00392). (V. A. Roman’kov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 157–173, March–April, 2009.     

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.     

Compact groups and fixed point sets

Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.

     

Topological dimensions for u-groups

We study some problems connected with algebraic geometry over a free metabelian group. We introduce the notions of topological dimensions which are based on the lengths of chains of irreducible closed sets, and study these dimensions.     

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G₁ ○ G₂ of two u-groups G₁ and G₂, and prove that G₁ ○ G₂ is also a u-group and that every u-group, which contains G₁ and G₂ as subgroups and is generated by these, is a homomorphic image of G₁ ○ G₂. It is stated that if G is a u-group then the coordinate group of an affine space Gⁿ is equal to G ○ F_n, where F_n is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005. Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education grant No. E00-1.0-12.     

On the fixing sets of dihedral groups

The fixing number of a graph $\Gamma$ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of $\Gamma$. The fixing set of a finite group $G$ is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to $G$. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group.     

Group rings with metabelian unit groups

Let F be a field of odd characteristic and G a group. In 1991 Shalev established necessary and sufficient conditions so that the unit group of the group ring FG is metabelian when G is finite. Here, in the modular case, we do the same without restrictions on G. In particular, new cases emerge when G contains elements of infinite order.     

New difference sets in nonabelian groups of order 100

In two groups of order 100 new difference sets are constructed. The existence of a difference set in one of them has not been known. The correspondence between a (100, 45, 20) symmetric design having regular automorphism group and a difference set with the same parameters has been used for the construction. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 424–434, 2001     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

Voronoi Diagrams of Real Algebraic Sets

A collection of n (possibly singular) semi-algebraic sets in ^d of dimension d–1, each defined by polynomials of maximal degree , has ((n) ^d ) first-order Voronoi cells (for any fixed d). In the nonhypersurface case, where the maximal dimension of the semi-algebraic sets is m d–2, the number of first-order Voronoi cells is bounded above by O(n ^{m+1 d} ) (for nonsingular semi-algebraic sets) or by O((n) ^d ) (in general). The complexity of the entire kth-order Voronoi diagram of a generic collection of n non-singular real algebraic sets in R ^d of maximal dimension m<d and maximal degree is O(n ^{min(d+k,2d)2(m+1)d} ).     

New partial difference sets in p‐groups

Latin square type partial difference sets (PDS) are known to exist in R × R for various abelian p‐groups R and in ?^t. We construct a family of Latin square type PDS in ?^t × ?^2nt_p using finite commutative chain rings. When t is odd, the ambient group of the PDS is not covered by any previous construction. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 394–402, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10029     

Separation of unitary representations of connected Lie groups by their moment sets

We show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.