The genus of a finitely generated nilpotent group G is defined as the set of isomorphism classes of finitely generated nilpotent groups K such that the p-localizations Kp, Gp are isomorphic for all primes p [19]. This notion turns out to be particularly relevant in the study of non-cancellation phenomena in group theory and homotopy theory. In the above definition, the restriction of finite gen...

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let Ext∗RG(M,M) be the cohomology ring associated to the RG-module M . Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element ζ ∈ Ext∗RG(M,M) is nilpotent (under Yoneda’s product) if and only if i...

Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F -split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or ...

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is ...

In this paper we determine the structure of (c,1,...,1) polynilpotent multiplier of certain class of groups. The method is based on the characterizing an explicit structure for the Baer invariant of a free nilpotent group with respect to the variety of polynilpotent groups of class row (c,1,...,1).

Introduction Gromov's Polynomial Growth Theorem [Gro81] characterizes the class of virtually nilpotent groups by their asymptotic geometry. Since Gromov's theorem it has been a major open question (see, e.g. [GH91]) to find an appropriate generalization for solvable groups. This paper gives the first step in that direction. One fundamental class of examples of finitely-generated solvable groups...