# Semistability of Graph Products

Research paper by **Michael Mihalik**

Indexed on: **24 Apr '20**Published on: **23 Apr '20**Published in: **arXiv - Mathematics - Group Theory**

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#### 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 $\infty$.