Geometric and homological methods in group theory: constructing small group resolutions

Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H. First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1). Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9). We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2.

Name dc:type.qualificationname

phd

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

London Metropolitan University

Year dc:date.issued

2011

Author dc:creator

• Gill, Olivia Jo

• 510 Mathematics

Dc Identifier Grantnumber

N/A

OAI identifier oai:identifier

oai:repository.londonmet.ac.uk:7533