We study almost contact metric structures on 5-dimensional nilpotent Lie algebras and investigate the class of left invariant almost contact metric structures on corresponding Lie groups. We determine certain classes that a five-dimensional nilpotent Lie group can not be equipped with.

This self-contained paper is part of a series [FF1, FF2] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante-Thurston proved that every nilpotent subgroup of Diff(S) is abelian. One of our main results is a sharp converse: Diff(S) contains every finitely-generated, torsion-free nilpotent group.

We introduce left and right series of semi-braces. This allows to define nilpotent study the structure such semi-braces generalize some results, known for skew braces, B that set additive idempotents E is an ideal . Finally we concept a semi-brace show multiplicative group nilpotent.

Let N be a nilpotent group normal in a group G. Suppose that G acts transitively upon the points of a finite non-Desarguesian projective plane P. We prove that, if P has square order, then N must act semi-regularly on P. In addition we prove that if a finite non-Desarguesian projective plane P admits more than one nilpotent group which is regular on the points of P then P has non-square order a...

A group is said to have the R∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R∞ property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n ≥ 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point ...

We describe the nilpotent subgroups of the group Bir(P(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k > 1; then either each element of N has finite order, or N is virtually metabelian.