Semistability of Graph Products

Abstract

A {\it graph product} $G$ on a graph $\Gamma$ is a group defined as follows: For each vertex $v$ of $\Gamma$ there is a corresponding non-trivial group $G_v$. The group $G$ is the quotient of the free product of the $G_v$ by the commutation relations $[G_v,G_w]=1$ for all adjacent $v$ and $w$ in $\Gamma$. A finitely presented group $G$ has {\it semistable fundamental group at $\infty$} if for some (equivalently any) finite connected CW-complex $X$ with $\pi_1(X)=G$, the universal cover $\tilde X$ of $X$ has the property that any two proper rays in $\tilde X$ are properly homotopic. The class of finitely presented groups with semistable fundamental group at $\infty$ is known to contain many other classes of groups, but it is a 40 year old question as to whether or not all finitely presented groups have semistable fundamental group at $\infty$. Our main theorem is a combination result. It states that if $G$ is a graph product on a finite graph $\Gamma$ and each vertex group is finitely presented, then $G$ has non-semistable fundamental group at $\infty$ if and only if there is a vertex $v$ of $\Gamma$ such that $G_v$ is not semistable, and the subgroup of $G$ generated by the vertex groups of vertices adjacent to $v$ is finite (equivalently $lk(v)$ is a complete graph and each vertex group of $lk(v)$ is finite). Hence if one knows which vertex groups of $G$ are not semistable and which are finite, then an elementary inspection of $\Gamma$ determines whether or not $G$ has semistable fundamental group at $\infty$.