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2019 Neighborhood complexes of Cayley graphs with generating set of size two
Jennifer R. Hughes
Rocky Mountain J. Math. 49(6): 1895-1907 (2019). DOI: 10.1216/RMJ-2019-49-6-1895

Abstract

For a group $G$ generated by $S\doteq \{g_1,\ldots ,g_n\}$, one can construct the Cayley graph $\mathrm {Cayley}({G},{S})$. Given a distance set $D\subset \mathbb Z _{\geq 0}$ and $\mathrm{Cayley}{G}{S}$, one can construct a $D$-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. 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 paper uses tools from group theory, representation theory and homological algebra to further our understanding of the interplay between generated groups, corresponding representations on their associated $D$-neighborhood complexes and the homology of the $D$-neighborhood complexes. This paper is an exposition of the results in my dissertation focusing on the case of two generators.

Citation

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Jennifer R. Hughes. "Neighborhood complexes of Cayley graphs with generating set of size two." Rocky Mountain J. Math. 49 (6) 1895 - 1907, 2019. https://doi.org/10.1216/RMJ-2019-49-6-1895

Information

Published: 2019
First available in Project Euclid: 3 November 2019

zbMATH: 07136585
MathSciNet: MR4027240
Digital Object Identifier: 10.1216/RMJ-2019-49-6-1895

Subjects:
Primary: 05E18
Secondary: \allowbreak 55U10, 05E45

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 6 • 2019
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