Graph Powers of Groups II: The RA Matrix

For a graph $\Gamma$ and group $G$, $G^\Gamma$ is the subgroup of $G^{|\Gamma|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $\Gamma$. There is a natural epimorphism $G^\Gamma \to (G/[G,G])^\Gamma$ with kernel $[G,G]^n \cap G^\Gamma$. When $[G,G]^n \leq G^\Gamma$, the structure of $G^\Gamma$ is easily described from $(G/[G,G])^\Gamma$. Fixing $\Gamma$, if $[G,G]^{|\Gamma|} \leq G^\Gamma$ for all $G$, we say that $\Gamma$ is RA (reducible to abelian). We showed in [2] that wide classes of graphs are RA, including graphs of girth 5 or more. The key tool is the RA matrix $C_{\Gamma}$, and we showed that $\Gamma$ is RA if and only if the row space $Row(C_\Gamma) = \mathbb Z^{|\Gamma|}$. Here, we study the possibilities for the elementary divisors of $C_\Gamma$; the more nontrivial elementary divisors we get, the further $\Gamma$ is from being RA (and the harder $G^\Gamma$ is to describe). We show that while many graphs, including those of girth 4, cartesian products, and most tensor products have at most one nontrivial elementary divisor, one can construct a graph of girth 3 with any prescribed set of elementary divisors and $\mathbb Z$-nullity.