Cayley complexes embedded in three dimensions

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This project is mainly in the area of algebraic Combinatorics, which the EPSRC is committed to continue supporting. But it also uses important machinery from Geometric Topology, and can be expected to have an impact there too.The student will first gain a thorough understanding of classical results of Geometric Topology about groups acting on spheres, as well as some modern results in graph theory. He will then try to extend these results from 2-dimensions to 3-dimensions: results about groups acting on the sphere need to be extended to group actions on the 3-dimensional sphere. Results about graphs embedded on a surface need to be extended into embeddings of 2-dimensional complexes into 3-dimensional space. Concretely, the project aims at proving the following theorem:A finite group acts by homeomorphisms on the 3-dimensional sphere if and only if it admits a Cayley complex embedded well in the 3-dimensional sphere.