Profinite Graphs

Abstract

This chapter contains the first notions about profinite graphs and their basic properties. A profinite graph is a graph in the usual sense, but it also has the structure of a topological space (compact, Hausdorff and totally disconnected). A finite graph with the discrete topology is a basic example of a profinite graph, and it is shown that every profinite graph can be expressed as an inverse limit of finite graphs. Most of the profinite graphs of interest in this book arise in connection with profinite groups (i.e., Galois groups), and they are usually infinite. One way of obtaining a profinite graph is by constructing the Cayley graph of a profinite group with respect to a compact subset.

Associated with a profinite graph there is a short sequence of profinite modules over a profinite ring, which depends on a class $\mathcal {C}$ of finite groups. Using such a sequence one develops the notion of ‘tree’ ($\mathcal {C}$-tree or $\pi$-tree, where $\pi$ is a set of prime numbers). It is proved that, for certain pseudovarieties $\mathcal {C}$, the Cayley graph of a free pro-$\mathcal {C}$ group is a $\mathcal {C}$-tree.

The chapter contains many examples that illustrate concepts and properties in profinite graphs.