Neighborhood complexes of Cayley graphs with generating set of size two

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.