GAP (Guarana) - Chapter 1: Introduction

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1 Introduction
1.1 About Guarana
1.2 Setup for computing the correspondence
1.3 Collection

### 1 Introduction

#### 1.1 About Guarana

In this package we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups; it is a correspondence that associates to every \(\Q\)-powered nilpotent group \(H\) a unique rational nilpotent Lie algebra \(L_H\) and vice-versa. The Mal'cev correspondence was discovered by Anatoly Mal'cev in 1951 [Mal51].

#### 1.2 Setup for computing the correspondence

Let \(G\) be a finitely generated torsion-free nilpotent group, i.e.\ a \(T\)-group. Then \(G\) can be embedded in a \(\Q\)-powered hull \(G^\). The group \(G^\) is a \(\Q\)-powered nilpotent group and is unique up to isomorphism. We denote the Lie algebra which corresponds to \(G^\) under the Mal'cev correspondence by \(L(G)= L_{G^}\). We provide an algorithm for setting up the Mal'cev correspondence between \(G^\) and the Lie algebra \(L(G)\). That is, if \(G\) is given by a polycyclic presentation with respect to a Mal'cev basis, then we can compute a structure constants table of \(L(G)\). Furthermore for a given \(g\in G\) we can compute the corresponding element in \(L(G)\) and vice versa.

#### 1.3 Collection

Every element of a polycyclically presented group has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Such an algorithm lies at the heart of most methods dealing with polycyclically presented groups. The current state of the art is collection from the left [Geb02][LGS90][VL90] }. This package contains a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection [AL07]. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory.

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