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Automorphisms of free products of groups

dc.contributorBrookes, Christopher
dc.creatorGriffin, James Thomas
dc.date.accessioned2018-11-24T23:26:13Z
dc.date.available2013-02-22T18:36:03Z
dc.date.available2018-11-24T23:26:13Z
dc.date.issued2013-02-05
dc.identifierhttp://www.dspace.cam.ac.uk/handle/1810/244265
dc.identifierhttps://www.repository.cam.ac.uk/handle/1810/244265
dc.identifier.urihttp://repository.aust.edu.ng/xmlui/handle/123456789/3787
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.
dc.languageen
dc.publisherUniversity of Cambridge
dc.publisherDepartment of Pure Mathematics and Mathematical Statistics
dc.rightshttp://creativecommons.org/licenses/by/2.0/uk/
dc.rightsAttribution 2.0 UK: England & Wales
dc.subjectAutomorphism groups
dc.subjectCohomology of groups
dc.titleAutomorphisms of free products of groups
dc.typeThesis


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