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Generalisations of groups and Cayley graphs
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Wendland, Alex (2019) Generalisations of groups and Cayley graphs. PhD thesis, University of Warwick.
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WRAP_Theses_Wendland_2019.pdf - Submitted Version - Requires a PDF viewer. Download (2259Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3717701~S15
Abstract
Groups are one of the most fundamental objects in Mathematics and have been generalised in many fashions. This thesis focuses on two generalisations.
Within the study of groups by geometric and combinatorial group theorists, instead of thinking about groups as an algebraic object they choose to study them through their Cayley graph. This has paved the way to many simplified proofs of properties about groups. Cayley graphs are vertex transitive graphs with a regular action by a group. However, not all vertex transitive graphs have a regular action and so cannot be Cayley graphs. This is reflected in the comparable levels of knowledge about them. The first chapter in this thesis generalises the concept of a group presentation and their associated Cayley graph. We hope this will open the door for techniques from combinatorial and geometric group theory to be applied to the study of vertex transitive groups.
The second is the study of groups in higher categories. Cayley pointed out that the study of groups is really just the study of symmetries. When we categorify groups into the setting of 2-categories, we study the symmetries between the symmetries given by a classical group. That is we allow the group axioms to hold only up to natural isomorphism. From this point of view we study 2-groups in the same way people studied classical groups, namely through their actions on vectors spaces. In this setting the 2-Vector spaces. The work provides an explicit formula for their characters.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Combinatorial group theory, Geometric group theory, Cayley graphs | ||||
Official Date: | 2019 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Georgakopoulos, Agelos | ||||
Format of File: | |||||
Extent: | vi, 122 leaves : illustrations (some colour) | ||||
Language: | eng |
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