Chapter 9: Fundamental Physics

Section 8: The Relationship of Space and Time

Cayley graphs

As discussed on page 938, the structure of a group can be represented by a Cayley graph where nodes correspond to elements in the group, and connections specify results of multiplying by generators. The transitivity of group multiplication implies that Cayley graphs always have the property of being symmetric (see above). The number of connections at each node is fixed, and given by the number of distinct generators and inverses. In cases such as the tetrahedral group A4 there are 3 connections at each node. The relations among the generators of a group can be thought of as constraints defining the Cayley graph. As mentioned on page 938, there are finite groups that have simple relations but at least very large Cayley graphs. For infinite groups, it is known (see page 938) that in most cases Cayley graphs are locally like trees, and so do not have finite dimension. It appears that only when the group is nilpotent (so that certain combinations of elements commute much as they do on a lattice) is there polynomial growth in the Cayley graph and thus finite dimension.

From Stephen Wolfram: A New Kind of Science [citation]