Polycyclic Groups Research Articles

Synthesis and biological evaluation of novel hydrazide based cytotoxic agents

Recently, we designed and synthesized a series of pyrroloquinoxaline compounds with hydrazine moiety linking a nitrogen-containing polycyclic group to a heteroaroyl system. Several derivatives, with attractive drug-like properties, were identified as promising cytotoxic agents, showing excellent potency in a panel of cancer cell lines. In the current study, we synthesized a further 19 new analogues to optimize their physicochemical properties and assess a coherent mechanism of action. Several chemical modifications were made to the reference compounds by varying the fused-ring system and/or the heteroacyl moiety. To evaluate their in vitro activity, we tested these compounds in six human cancer cell lines derived from different origins. Among them, two compounds ( and ) showed similar potency as the reference compounds with IC50 values in the sub-micromolar range in all cell lines tested. Furthermore, compound showed excellent in vivo efficacy in our preliminary human ovarian cancer mouse xenograft studies. Flow cytometric studies indicated that both derivatives interrupted cell cycle progression in colorectal cancer HCT116 cell lines and ovarian cancer SKOV-3 cells. Further mechanistic studies revealed that and were able to induce reactive oxygen species in SKOV-3 cells with apparently different kinetic patterns. Considering their cytotoxicity profiles in a variety of in vitro and in vivo cancer models, these hydrazide based compounds seem to have considerable potentials as novel chemotherapeutics.

The twisted conjugacy problem for endomorphisms of metabelian groups

Let M be a finitely generated metabelian group explicitly presented in a variety $ {\mathcal{A}}^2 $ 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).

Electrochemical and Fluorescent Properties of Ferrocenyl Chalcone with N-Ethyl Carbazole Group

These systems were designed in order to modulate the electrochemical and optical properties of the polycyclic moieties by the electron- donating ferrocenyl group through the enone linkage. However, the intrinsic optical properties of the naphthalene and anthra-cene moieties were quenched by the ferrocenyl moiety which acts as a quencher. In an attempt to achieve a breakthrough with this system, we adopted a carbazole derivative as the polycyclic group in ferrocenyl chalcones. Carbazole (C

Computing the nonabelian tensor squares of polycyclic groups

In this paper we develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups. This theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group.

Decision and Search in Non-Abelian Cramer-Shoup Public Key Cryptosystem

A method for non-abelian Cramer-Shoup cryptosystem is presented. The role of decision and search is explored, and the platform of solvable / polycyclic group is suggested. In the process we review recent progress in non-abelian cryptography and post some open problems that naturally arise from this path of research.

Generation of polycyclic groups

We give a new and self-contained proof of a theorem of Linnell and Warhurst that d(G) – d(Ĝ) ⩽ 1 for finitely generated virtually torsion-free soluble minimax groups G. We also give a simple sufficient condition for the equality d(G) = d(Ĝ) to hold when G is virtually abelian.

Some examples of Tychonoff groups

In this paper we survey the Tychono. property for discrete groups. General results and relations with amenability, with discrete potential theory (random walks), and with the theory of weights and of characters are discussed. We then present a characterization of Tychonoff property for virtually abelian groups, for polycyclic groups, and for linear groups. We also consider the case of metabelian groups, and several examples are given.

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.

Extensions of lattice-ordered groups

A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension.

Stable cohomology of compact homogeneous spaces

The cohomology of certain compact homogeneous spaces is studied. The notion of stable cohomology (invariant under the passage to a finite covering) is introduced; examples of the calculation of this cohomology (Theorem 1) and its application to the study of the structure of compact homogeneous spaces (Theorem 2) are given. Several conjectures about properties of stable cohomology related to various areas of mathematics (such as topology and the cohomology of discrete (in particular, polycyclic) groups) are stated.

On the Zeroeth Complete Cohomology of Certain Polycyclic Groups

The zeroeth complete cohomology of a group G with coefficients in , is known to carry much information about the group. Its computation is rather complicated. Kropholler (1993b) and Benson and Kropholler (1995) stated that even when G is a polycyclic by finite group, no general formula is known for . Here we construct a projective resolution for a class of polycyclic groups and obtain information about , which indicates that no general formula for can be expected. Moreover, some groups in our class are counterexamples to a conjecture of Brown, which states that the least common multiple of the orders of the finite subgroups of a group of finite virtual cohomological dimension annihilates the complete cohomology of G. The first counterexample to this conjecture was given by Adem (1989) and more were obtained later by Adem and Carlson (1990) and Leary (1996). We obtain counterexamples that are not in those classes.

Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group

We describe effective algorithms for computing a polycyclic presentation of the Schur multiplicator, a Schur cover, the nonabelian exterior square and the nonabelian tensor square of a polycyclic group given by a polycyclic presentation. Additionally, we introduce a method to check whether a polycyclic group is capable.

Heterocarbenoids of germanium and tin and their polyhedral oxidation products: The case for thermodynamic product control in Group 14 chalcogenides

The polycyclic Group 14 amides [P(μ-N t Bu) 2P( t BuN) 2]M, M = Ge ( 4), Sn ( 5) were synthesized from cis-[P(μ-N t Bu) 2P( t BuNLi · THF) 2] and GeCl 2 · dioxane or SnCl 2, respectively. Oxidation of these heterocarbenoids or of the analogous diazastannylene [MeSi(μ- t BuN) 2SiMe( t BuN) 2]Sn with O 2, S 8 and Se n furnished the chalcogenides {[P(μ-N t Bu) 2P( t BuN) 2]GeO} 2 ( 6), {[P(μ-N t Bu) 2P( t BuN) 2]SnE} 2, E = O ( 7), S ( 8), Se ( 9), {[SP(μ-N t Bu) 2P( t BuN) 2]SnS} 2 ( 10), and {[MeSi(μ- t BuN) 2SiMe( t BuN) 2]SnE} 2, E = S ( 11), Se ( 12), respectively. All products ( 6– 12) were shown by single-crystal X-ray methods to consist of dimeric molecules with central (M–E) 2 rings, M = Group 14 element, E = chalcogen. The exclusive formation of dimeric compounds with bridging M–E–M bonds, vs. alternative monomeric structures with terminal M E bonds, is rationalized in terms of the thermodynamic favorability of the dimers. The case is made that most, if not all, currently known Group 14 chalcogenides, even those labeled “kinetically stabilized”, are really thermodynamic products.

BREADTH IN POLYCYCLIC GROUPS

The breadth of a polycyclic group is the maximum of h(G) - h(CG(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

A finitely-presented solvable group with a small quasi-isometry group

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large-scale geometry of lamplighter groups.

Using the Mal'cev correspondence for collection in polycyclic groups

We describe several approaches for realizing the Mal'cev correspondence between Q -powered nilpotent groups and nilpotent Lie algebras over Q . We apply it to fast collection in polycyclic groups. Our methods are fully implemented and publicly available. We report on the implementation and give runtimes for some example groups.

The non-abelian tensor product of polycyclic groups is polycyclic

The non-abelian tensor product of polycyclic groups is polycyclic

Erratum to: Continuous control and the algebraic L-theory assembly map

It was claimed in [Rosenthal D.: Continuous control and the algebraic L-theory assembly map. Forum Math. 18 (2006), 193–209] that virtually polycyclic groups admit a universal space for proper actions that satisfy the geometric assumptions of the main theorem there. However, it is unknown in general if such a space exists for these groups. In order to prove that the given assembly map for virtually polycyclic groups is a split injection, one can use [Bartels A. and Rosenthal D.: On the K-theory of groups with finite asymptotic dimension. Preprint 2006 arXiv:math.KT/0605088 v2].

Quasi-isometries and Rigidity of Solvable Groups

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as coarse differentiation.