The influence of the complete nonexterior square graph on some infinite groups

The notion of nonabelian exterior square may be formulated for a pro-$p$-group $G$ ($p$ prime), getting the complete nonabelian exterior square $G \widehat{\wedge} G$ of $G$. We introduce the complete nonexterior square graph $\widehat{\Gamma}_G$ of $G$, investigating finiteness conditions on $G$ from restrictions on $\widehat{\Gamma}_G$ and viceversa. This graph has the set of vertices $G − \widehat{Z}(G)$, where $\widehat{Z}(G)$ is the set of all elements of $G$ commuting with respect to the operator $\wedge$ , and two vertices $x$ and $y$ are joined by an edge if $x \widehat{\wedge} y \neq 1$. Studying $\widehat{\Gamma}_G$ , we find the well-known noncommuting graph as a subgraph. Moreover, we show results on the structure of $G$ and introduce a new class of groups, which originates naturally when $G$ is infinite but $\widehat{\Gamma}_G$ is finite.