Group action on neighborhood complexes of Cayley graphs
Date
2014
Authors
Hughes, Justin, author
Hulpke, Alexander, advisor
Peterson, Chris, advisor
Berger, Bruce, committee member
Cavalieri, Renzo, committee member
Wilson, James, committee member
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Abstract
Given G a group generated by S ā {g1, ā¦, gn}, one can construct the Cayley Graph Cayley (G,S). Given a distance set D ā Zā„0 and Cayley (G,S) one can construct a D-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. The group G acts on this chain complex and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of G. This thesis uses tools from group theory, representation theory, homo-logical algebra, and topology to further our understanding of the interplay between generated groups (i.e. a group together with a set of generators), corresponding representations on their associated D-neighborhood complexes, and the homology of the D-neighborhood complexes.
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Subject
neighborhood complexes
Cayley graphs
group actions