A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. U...

Theorem 1 gives an explicit formula for the heat kernel on an H -type group. Folland (2] has shown that for stratified nilpotent Lie groups the heat semigroup is a semigroup of kernel operators on LP, 1 5 p < oo and on Co. Cygan (1] has obtained formulas for heat kernels for any two step nilpotent simply connected Lie group. Cygan found the heat kernel for a free simply connected two step nilpo...

Abstract Shalom and Tao showed that a polynomial upper bound on the size of single, large enough ball in Cayley graph implies underlying group has nilpotent subgroup with index degree growth both bounded effectively. The third fourth authors proved optimal this subgroup, at expense making some other parts result ineffective. In present paper, we prove without any losses elsewhere. As consequenc...

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable ...

The concept of dominion (in the sense of Isbell) is investigated in several varieties of nilpotent groups. A complete description of dominions in the variety of nilpotent groups of class at most 2 is given, and used to prove nontriviality of dominions in the variety of nilpotent groups of class at most c for any c>1 . Some subvarieties of N2 , and the variety of all nilpotent groups of class at...

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...

Introduction. A long-standing problem in group theory is to determine the number of non-isomorphic groups of a given order. The inverse problem–determining the orders for which there are a given number of groups–has received considerably less attention. In this note, we will give a characterization of those positive integers n for which there exist exactly 2 distinct groups of order n (up to is...

We shall say that an automorphism a is nilpotent or acts nilpotently on a group G if in the holomorph H= [G](a) of G with a, a is a bounded left Engel element, that is, [H, ¿a] = l for some natural number ¿. Here [H, ka] means [H, (k — l)a] with [H, Oa] denoting H. Let G' denote the commutator subgroup [G, G], and let $(G) denote the Frattini subgroup of G. If a is an automorphism of a nilpoten...

Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a Gmodule. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. Thesis Supervisor: David Alexander Vogan Title: Professor of Mathematics

Combinatorial properties of zeons have been applied to graph enumeration problems, colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series elementary zeon functions are naturally reduced finite sums by virtue the nilpotent zeons. Further, extension any analytic complex function has polynomial representations on as...