 
Summary: ON THE CAYLEY GRAPH OF A GENERIC FINITELY
PRESENTED GROUP.
G. N. ARZHANTSEVA AND P.A. CHERIX
Abstract. We prove that in a certain statistical sense the Cayley graph of
almost every finitely presented group with m – 2 generators contains a subdi
vision of the complete graph on l Ÿ 2m+ 1 vertices. In particular, this Cayley
graph is non planar. We also show that some group constructions preserve the
planarity.
1. Introduction
To any finite presentation of a group in terms of generators and defining relations
there is an associated Cayley graph. This graph depends on the choice of the group
generating set. So, in general, the same group has completely different Cayley
graphs (from the graph theory viewpoint). In particular, it is not hard to find a
group and two different sets of generators such that the Cayley graph with respect
to one generating set is planar and not planar with respect to the other. As an
example, take the cyclic group Z=5Zof order five and two generating sets. The
first one consisting of a single non trivial element and the other one consisting of
all non trivial elements. Then with respect to the first generating set, the Cayley
graph is a cycle, so it is planar, but with respect to the second one it is not, as it
is the complete graph on five vertices.
