In this paper, we prove results concerning the large scale geometry of connected, simply connected nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT0 metric space or an Alexandrov metric space. The main technical aspect of this work is the proof of a limited...

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying π0 is the theory of ordinary n-nilpotent groups and that t...

In 1971, Eggert [2] conjectured that for a finite commutative nilpotent algebra A over a field K of prime characteristic p > 0, dimA ≥ p dimA(p), where A(p) is the subalgebra of A generated by all the elements xp, x ∈ A and dimA, dimA(p) denote the dimensions of A and A(p) as vector spaces over K. In [3], Stack conjectures that dimA ≥ p dimA(p) is true for every finite dimensional nilpotent alg...

Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup CG(H) of the complement is nilpotent of class c. It is proved that G has a nilpotent characteristic subgroup of index bounded in terms of c, |CG(F )|, and |F | whose nilpotency class is bounded in terms of c and |H| only. This gener...

Abstract. Let Bn be the group of upper-triangular invertible n×n matrices and Xn be the variety of strictly upper triangular n × n matrices of nilpotent order 2. Bn acts on Xn by conjugation. In this paper we describe geometry of orbits Xn/Bn in terms of link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotent order 2 and intersectio...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it virtually nilpotent. However, until now the only examples known were abelian. In this note we furnish an example 2-step nilpotent having certain set.

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

We adapt the p-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most 9 over F2 and those of dimension at most 7 over F3 and F5 .