Show simple item record

dc.contributor.advisorBrookes, Christopher
dc.contributor.authorGriffin, James Thomas
dc.description.abstractThe symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.en_GB
dc.rightsAttribution 2.0 UK: England & Walesen
dc.subjectAutomorphism groupsen_GB
dc.subjectCohomology of groupsen_GB
dc.titleAutomorphisms of free products of groupsen_GB
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridgeen_GB
dc.publisher.departmentDepartment of Pure Mathematics and Mathematical Statisticsen_GB

Files in this item


This item appears in the following Collection(s)

Show simple item record

Attribution 2.0 UK: England & Wales
Except where otherwise noted, this item's licence is described as Attribution 2.0 UK: England & Wales