Using the Mal’cev correspondence for collection in polycyclic groups

Applications of Lie Methods to Computations with Polycyclic Groups

In this thesis we demonstrate the algorithmic usefulness of the so-called Mal’cev correspondence for computations with infinite polycyclic groups. This correspondence between Q-powered nilpotent groups and rational nilpotent Lie algebras was discovered by Anatoly Mal’cev in 1951. We show how the Mal’cev correspondence can be realized on a computer. We explore two possibilities for this purpose ...

Algorithmic Use of the Mal’cev Correspondence

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically pre...

The Mal’cev correspondence for counting problems in groups and rings

On the Irreducibility of Pseudovarieties of Semigroups

We show that, for every pseudovariety of groups H, the pseudovariety H̄, consisting of all finite semigroups all of whose subgroups lie in H, is irreducible for join and the Mal’cev and semidirect products. The proof involves a Rees matrix construction which motivates the study of iterated Mal’cev products with the pseudovariety of bands. We further provide a strict infinite filtration for H̄ usi...

Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal’cev correspondence, these results have corollaries pertaining to ze...