Title:

Automorphisms of graph products of groups

Let r be a graph with vertex set V and for every v E V let Gv be a group with
present ation (Sv I Rv). Let E ~ V X V be the set of pairs of adj acent vertices. Then we
define the group G = Gr to be the group with presentation
G = (SvVv E VI R; Vv E V, [Sv, SVI] = 1 iff (v,v') E E).
In [2, LEMMA 3.3] it is shown that up to isomorphism G is independent of the choice
of presentation of each group Gv. We call the group G a graph product of groups.
Graph products include as special cases free products and direct products, corresponding
to the graph G being dixcrete and complete respectively. If the vertex groups G; are
infinite cyclic then G is called a graph group and we identify each vertex v with a fixed
generator of the vertex group Gv•
There is a normal form theorem for graph products which is a generalisation of the
normal form theorem for free products and which was proved in [2]. In Part 1we give an
alternative proof. We then move on to the study of automorphisms of graph products. In
full generality this is an impossible task; however some progress can be made in certain
special cases. We first consider the case where G is a graph group. Servatius in [1] gave
a simple set of elements of Aut( G), which he calls elementary automorphisms, and proved
that if certain conditions are imposed on the graph r then the elementary automorphisms
generate Aut(G). In Part 2 we will prove that this holds for all finite graphs r.
In Part 3 we study Aut( G) in the case where each vertex group Gv is cyclic of order p
for some fixed prime p and we find a simple set of generators for Aut(G). In the case p = 2
we also obtain a presentation for Aut( G). In this case G is a rightangled Coxeter group
