Geometric equivalence of nilpotent groups

Nilpotent, torsion free groups are considered. Sufficient conditions are presented for a nilpotent, torsion free group to be geometrically equivalent to its Mal’tsev completion. Also some results are achieved in describing the classes of geometric equivalence of class 2 nilpotent, torsion free groups with center of small rank. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 259–270.     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.     

Automorphisms of groups and of schemes of finite type

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS _n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.” Partially supported by the NSF under Grant MCS 80-05802.     

Five dimensional Bieberbach groups with trivial centre

Let Γ be a Bieberbach group—that is a torsion free crystallographic group. In this paper is given a list of the isomorphy types of all holonomy groups of five-dimensional Bieberbach groups with trivial centre.     

On distinguished local coordinates for locally homogeneous affine surfaces

We give a new short self-contained proof of the result of Opozda (Differ Geom Appl 21:173–198, 2004) classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski (Monatsh Math 153:1–18, 2008). Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken previously in the literature.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

Holonomy groups of five dimensional Bieberbach groups

Let Γ be a Bieberbach group—that is a torsion free crystallographic group. In this paper we give a list of the isomorphism types of all holonomy groups of five-dimensional Bieberbach groups. An application to the problem of estimating the covolume of a discrete group of orientation-preserving isometries of hyperbolic 6-space is also given. This work was supported by Polish grant nr.0627/P3/93/04     

On holomorphic sections in Teichmüller spaces

It is proved that, for any elementary torsion free Fuchsian group F, the natural projection from the Teichmiiller curve V（F） to the Teichmiiller space T（F） has no holomorphic section.     

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on T³ and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

Existential questions in (relatively) hyperbolic groups

We study the decidability of the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. Our tools are Rips and Sela’s canonical representatives for these groups, and solvability of equations with rational constraints (involving finite state automata) in free groups and free products.     

Torsion free extensions and perfect localizations

This note may be viewed as a continuation of the ideas developed in [13]. We start by reviewing some principles concering bimodule localization, the fundamentals of which the reader is assumed to be familiar with. In particular, the main properties of perfect localization and their relation to extensions are recollected. Next, we introduce the idea of a torsion free extension and prove its main features, especially in relation to torsion theories. We conclude by generalizing a theorem of L. Rowen's [9] concerning rings which are torsion free over their center to arbitrary torsion free extensions. His results are thus shown to be of torsion theoretic nature.     

Gruppo delle terne quasi pitagoriche primitive

We study the groupG _m of primitive solution of the diophantine equationx ²+my²=z² (m>1, squarefree). Form∈3 this group is torsion free, form=3 it has a torsion element of order 3; moreover for a finite number of values ofm we prove thatG _m is a direct sum of infinite cyclic groups and we give the generators ofG _m in terms of the primes represented by the quadratic forms of discriminant Δ=−4m.      

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.     

Homology of free Lie powers and torsion in groups

LetG be a group that is given by a free presentationG=F/R, and let_γ4 R denote the fourth term of the lower central series of R. We show that ifG has no elements of order 2, then the torsion subgroup of the free central extensionF/[_γ4 R,F] can be identified with the homology groupR _γ6(G, ℤ/2ℤ). This is a consequence of our main result which refers to the homology ofG with coefficients in Lie powers of relation modules.     

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ?. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.     

An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space

We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike principal normal in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at points where the curve experiences high torsion. In our construction the tubular surface’s twist is bounded by the spacelike curve’s curvature and is independent of the spacelike curve’s torsion.      

Rank-1 quotient divisible groups

An Abelian group is called quotient divisible if it does not contain nonzero torsion divisible subgroups but does contain a free finite-rank subgroup such that the quotient group by it is divisible. In this paper, we will describe rank-1 quotient divisible groups with the help of cocharacteristics, and we will describe the endomorphisms of these groups as well. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 25–33, 2007.     

The metric anomaly of analytic torsion at the boundary of an even dimensional cone

The formula for analytic torsion of a cone in even dimensions is composed of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The third “singular” contribution is an intricate spectral invariant of the cone base. We identify the third term as the metric anomaly of the analytic torsion coming from the non-product structure of the cone at its regular boundary. Hereby we filter out the actual contribution of the concial singularity and identify the analytic torsion of an even-dimensional cone purely in terms of the Betti numbers and the analytic torsion of the cone base.