Semistability of graph products

Abstract

A graph product G on a graph Γ is a group defined as follows: For each vertex v of Γ there is a corresponding non-trivial group Gv . The group G is the quotient of the free product of the Gv by the commutation relations for all adjacent v and w in Γ. Semistability at is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at . While many classes of groups are known to contain only semistable at groups, there are only a few combination results for such groups. Our main theorem is such a result. It states that if G is a graph product on a finite graph Γ and each vertex group is finitely presented, then G has non-semistable fundamental group at if and only if there is a vertex v of Γ such that Gv 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 Γ determines whether or not G has semistable fundamental group at .