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 right-angled Coxeter group
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:412581 |
| Date | January 1992 |
| Creators | Laurence, Michael Rupen |
| Publisher | Queen Mary, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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