Calculating the mislin genus for a certain family of nilpotent groups

It is known that the Mislin genus of a finitely generated nilpotent group N with finite commutator subgroup admits an abelian group structure. In this paper, we compute explicitly that structure under the following additional assumptions: The torsion subgroup TN is abelian, the epimorphism N→N/TN splits and all automorphisms of TN commute with cinjugation by elements of N. Among the groups satisfying these conditions are all nilpotent split extensions of a finite cyclic group by a finitely free abelian group. We further prove that the function M ? M × N^k­1 k ≥ 2, which is in general a surjective homomorphism from the genus of N onto the genus of N^k , is an isomorphism at least in an imporatnt special case. Applications to the study of non-cancellation phenomena in group theory are given.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

About a special class of two-dimensional complex Finsler spaces

In this paper we investigate a class of two-dimensional complex Finsler spaces, called η-Einstein, in [3]. The holomorphic sectional curvatures of such spaces, in directions of the local complex Berwald frames {l, m, [`(l)],[`(m)]\bar l,\bar m} and {λ, μ, [`(l)] ,[`(m)]\bar \lambda ,\bar \mu } are studied. We classify some subclasses of η-Einstein spaces, with respect to the horizontal holomorphic sectional curvature in direction λ. Finally, a special approach is devoted to the holomorphic bisectional curvatures of the two — dimensional η-Einstein spaces.     

Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

Rationally equivalent nilpotent groups and spaces

Examples are constructed of rationally isomorphic, finitely generated nilpotent groups and such that there are no homomorphisms inducing rational isomorphisms. Similar examples are constructed of nilpotent, or even simply-connected, finite CW-complexes.

     

Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G₁,G₂ is constructed with the properties that G₁ and G₂ are p-isomorphic for all primes p, yet Aut(G₁) and Aut(G₂) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.     

On a Theorem of Mislin

An alternative proof is given of a result, originally due toGuido Mislin, giving necessary and sufficient conditions forthe inclusion of a subgroup to induce an isomorphism in modp cohomology.     

On certain linear spaces of nilpotent matrices

In this note we describe explicitly those linear spaces of nilpotent matrices, which are stable under conjugation by diagonal matrices. As an application, we deduce a generalized version of a result of Valcher and Fornasini on pairs of nonnegative nilpotent matrices.