13 - Finite unramified coverings and Galois groups

Summary

In this chapter we begin the study of Galois theory for finite unramified covering graphs. It leads to a generalization of Cayley and Schreier graphs and provides factorizations of zeta functions of normal coverings into products of Artin L-functions associated with representations of the Galois group of the covering. Coverings can also be used in constructions of Ramanujan graphs and in constructions of pairs of graphs that are isospectral but not isomorphic. Most of this chapter is taken from Stark and Terras [120]. Other references are Sunada [128] and Hashimoto [51]. Another theory of graph covering which is essentially equivalent can be found in Gross and Tucker [47]. Our coverings, however, differ in that we require all our graphs to be connected and in that our aim is to find analogs of the basic properties of finite-degree extensions of algebraic number fields and their zeta functions. It is also possible to consider infinite coverings such as the universal covering tree T of a finite graph X. We will not do so here except in passing. This is mostly a book about finite graphs, after all.

Definitions

If our graphs had no multiple edges or loops, our definition of covering would be Definition 2.9. If we want to prove the fundamental theorem of Galois theory for graphs with loops and multiple edges, Definition 2.9 will not be sufficient. We need to produce a more complicated definition of graph covering involving neighborhoods in directed graphs.