arXiv subjects
Publications and source records attributed to Daniel E. Cohen.
Let $Γ$ be a finite graph together with a group $G_v$ at each vertex $v$. The graph product $G(Γ)$ is obtained from the free product of all $G_v$ by factoring out by the normal subgroup generated by $\{g^{-1}h^{-1}gh; g\in G_v, h\in G_w\}$ for all adjacent $v,w$. In this note we construct a projective resolution for $G(Γ)$ given projective resolutions for each $G_v$, and obtain some applications.
Let $Γ$ be a finite graph, and for each vertex $i$ let $G_i$ be a finitely presented group. Let $G$ be the graph product of the $G_i$. That is, $G$ is the group obtained from the free product of the $G_i$ by factoring out by the smallest normal subgroup containing all $[g,h]$ where $g\in G_i$ and $h\in G_j$ and there is an edge joining i and j . We show that $G$ has an isoperimetric function of degree $k\ge 2$ (or an exponential isoperimetric function) if each vertex group has such an isoperimetric function.