Ordering Graph Products of Groups

Let Γ be a finite simple graph with vertex set V , so the edges may be taken to be unordered pairs of distinct elements of V . Assume that, for every v ∈ V , there is assigned a group Gv. The graph product GΓ is the quotient of the free product ∗v∈V Gv by the normal subgroup generated by all [Gu,Gv] for which {u,v} is an edge. (As usual, [Gu,Gv] means the subgroup generated by { x−1y−1xy | x ∈ Gu, y ∈ Gv } .) These groups were studied by E. R. Green in her thesis [6], and have since attracted considerable attention (see, for example, [7] and the references cited there). In this paper, it is shown that, if all Gv are right orderable, then GΓ is right orderable, and if all Gv are (two-sided) orderable then GΓ is orderable. Recall that a right order on a group G is a total order ≤ such that a≤ b and c ∈ G implies ac≤ bc. The strictly positive cone is the set P := {g ∈ G | 1 < g}. It has the properties PP⊆ P, G\{1}= P∪P−1 and P∩P−1 = / 0.