We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales-Jewett theorem ([BL2]) and contains nilpotent extensions of many known “abelian” results as special cases. 0. Introduction 0.1. The celebrated van der Waerden theorem on arithmetic progression...

We study the stability of certain spectra under some algebraic conditions weaker than commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra $$\mathcal {A},$$ prove that if $$x \in \hbox {comm}_{w}(a)$$ is nilpotent, then $$\sigma (x)=\sigma (x+a).$$ Among other things, {comm}(xy)\cup {comm}(yx)$$ or $$y\in for all $$x,y \m...

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0; x1; : : : ; xN 2 G, N 2 N, then one can always …nd a uniformly computably enumerable (i.e. uniformly 1) ascendant sequence of order type ! + 1 of subgroups in G beginning with hx0; x1; : : : ; xN iG, the subgroup generated by x0; x1; : : : ; xN in G. Th...

This article examines aspects of the theory of locally nilpotent linear groups. We also present a new classification result for locally nilpotent linear groups over an arbitrary field F. 1. Why Locally Nilpotent Linear Groups? Linear (matrix) groups are a commonly used concrete representation of groups. The first investigations of linear groups were undertaken in the second half of the 19th cen...

Abstract In this article, some new sufficient conditions of p -nilpotency finite groups are obtained by using c -normality and Φ-supplementary the maximal or 2-maximal subgroups Sylow -subgroups.

We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the pro-p topology on each of the factors.

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the “order formula” of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebr...

It has been known for at least thirty years that certain nilpotent groups cannot be Kähler groups, i.e., fundamental groups of compact Kähler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-ste...

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups. Section

In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pai...