Nilpotent Quotients of Associative -Algebras and Augmentation Quotients of Baumslag-Solitar Groups

Abstract

We describe the functionality of the package zalgs for the computer algebra system GAP. The package contains an implementation of the nilpotent quotient algorithm for finitely presented associative -algebras described in [3]. As an application of this algorithm we calculate augmentation quotients, i.e. successive quotients of powers of the augmentation ideal I(G) of the integral group ring , where G is a finitely presented group. We apply these methods to obtain conjectures for augmentation quotients of the Baumslag-Solitar groups BS(m, n) with equal to 0, 1 or a prime p.

Introduction

An associative -algebra A is called nilpotent of class if its series of power ideals has the form . The power ideal , is the ideal in A generated by all products of length i. In [3] we introduced so called nilpotent presentations to describe such algebras in a way that exhibits their nilpotent structure. We also introduced a nilpotent quotient algorithm, which computes a nilpotent presentation for the class-c quotient for a given finitely presented associative -algebra A and a non-negative integer c. An implementation of this algorithm is available in the package zalgs [5] for the computer algebra system GAP [4].

The purpose of this paper is to describe the functionality of the zalgs package and to exhibit applications of the nilpotent quotient algorithm. In particular, we apply the algorithm in the calculation of augmentation quotients, i.e. the quotients , where G is a finitely presented group and I(G) denotes the augmentation ideal of the integral group ring . The augmentation ideal I(G) is defined as the kernel of the augmentation map

where and . The augmentation quotients are interesting objects studied in the integral representation theory of groups. We present conjectures on the augmentation quotients of certain Baumslag-Solitar groups, which are based on computer experiments using the zalgs package.

Nilpotent Presentations and Nilpotent Quotient Systems

As all algebras considered in this paper will be associative -algebras, we will simply refer to them as algebras. For completeness we recall several important definitions from [3].

Definition 1

Let A be a finitely generated algebra of class c and . We call a weighted generating sequence for A with powers and weights if

1. , i.e. A is the -span of .

2. for .

3. is minimal in with respect to the property that , or , if such an does not exist.

4. The elements with generate for .

Definition 2

A consistent weighted nilpotent presentation for a finitely generated nilpotent algebra A is given by a weighted generating sequence with powers , weights and relations of the following form:

1. for all where .

2. for and .

3. The and are integers with if .

We note that every finitely generated nilpotent algebra has a consistent weighted nilpotent presentation, see [3, Theorem 7]. In an algebra A given by a consistent weighted nilpotent presentation, we can determine a normal form for each , i.e. there are uniquely determined with

and if .

Definition 3

Let be a finitely presented algebra, and let be the natural homomorphism. A nilpotent quotient system describes using the following data:

1. A consistent weighted nilpotent presentation for with generators , powers , weights , multiplication relations for and power relations if .

2. Images for given in normal form.

3. Definitions , with being an integer or a pair of integers, s.t.

   • If is an integer, then and .
   • If , then , where and .

The description of using this data is very useful for computational purposes and usually the output of our calculations will be in the form of nilpotent quotient systems. The following example shall illustrate the definition of nilpotent quotient systems.

Example 1

Consider the finitely presented algebra given by

Then a nilpotent quotient system for consists of:

• generators with powers (2, 0) and weights (1, 1),

• the power relation and the multiplication relations for ,

• images and , and

• definitions (1, 2).

A nilpotent quotient system for consists of:

• generators with powers (2, 0, 2, 2) and weights (1, 1, 2, 2),

• the power relations and the multiplication relations , , and for all other ,

• images and , and

• definitions (1, 2, (1, 2), (2, 1)).

Functionality

The central functionality provided by the zalgs package is the function

NilpotentQuotientFpZAlgebra(A, c),

which takes as input a finitely presented associative -algebra A and a non-negative integer c. The output is a nilpotent quotient system for .

Example 2

The following is an example calculation of a nilpotent quotient system in GAP for the class-2 quotient of the associative -algebra considered in Example 1 above, i.e.

To carry out the computation, we start by defining A as the quotient of the free associative -algebra on two generators by the given relations.

We then call NilpotentQuotientFpZAlgebra(A, 2) to compute the class-2 quotient. The output contains lists for the definitions dfs, the powers pows, the weights wgs and an integer dim indicating the dimension of the quotient. The entries for the images imgs, power relations ptab and multiplication relations mtab are to be interpreted as coefficients of normal forms. For computational purposes there is an additional entry rels in the output.

In [3, Section 5], we describe how to obtain a presentation P for an algebra, such that is isomorphic to the class-c quotient of P. The nilpotent quotient algorithm can now be applied to determine this nilpotent quotient. The following functions are available to calculate the class-c quotient of the augmentation ideal of integral group rings.

AugmentationQuotientFpGroup(G, c),

AugmentationQuotientPcpGroup(G, c),

which take as input a finitely presented group or a polycyclically presented group G, respectively, and a non-negative integer c. The output in both cases is a nilpotent quotient system for . Note that the augmentation quotients for can be read off from this.

Augmentation Quotients of Baumslag-Solitar Groups

In [3, Section 5], we describe how to obtain, for a given finitely presented group G, a presentation for an algebra A, such that is isomorphic to the class-c quotient of A. We apply these methods to compute augmentation quotients of the Baumslag-Solitar groups BS(m, n), which for are given by the presentations

These one-relator groups form an interesting set of groups with applications in combinatorial and geometric group theory, e.g. the group BS(1, 1) is the free abelian group on two generators and arises as the fundamental group of the Klein bottle. The Baumslag-Solitar groups were introduced in [2] as examples of non-Hopfian groups and the isomorphism problem for these groups has been considered in [6].

We carried out computer experiments to gain some insight into the structure of the augmentation quotients for and small values of k. Our computations suggest the following conjectures for certain special cases:

Conjecture 1

Let p be a prime.

1. If and , then

   

   where

   

   are the Eulerian numbers and the values C(k) are given by the recursion

   

   where B(u) is the number of 8-element subsets of whose elements sum to a triangular number, i.e. a number of the form , .
2. If , then for all :

   
3. If , then for all :

   

   where is the k-th triangular number.

The behaviour appears to be more complicated if contains several (not necessarily distinct) prime factors, as is illustrated in the following example.

Example 3

Let G be the Baumslag-Solitar group . Then the first few augmentation quotients are as follows:

Further Aims

Bachmann and Grünenfelder [1] showed that for finite groups G the sequence for is virtually periodic, i.e. there exist and such that for all . It will be interesting to extend our methods to allow the determination of these parameters, which in theory allows to determine all augmentation quotients for a given finite group G.

Furthermore, we plan to extend our algorithms to compute nilpotent presentations for the largest associative -algebra on d generators so that every element a of the algebra satisfies , i.e. compute -algebra analogues of Burnside groups and Kurosh algebras.