Cofinite Graphs and their Profinite Completions

TL;DR

This paper extends the concept of cofinite groups to graphs, defining cofinite graphs and their uniform completions, and proves the existence and uniqueness of such completions.

Contribution

It introduces a framework for cofinite graphs with uniform structures and establishes their unique completions, generalizing prior work on cofinite groups.

Findings

Defined cofinite graphs with uniform structures

Proved existence of unique cofinite completions

Generalized cofinite group concepts to graphs

Abstract

We generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions. The idea of constructing a cofinite graph starts with defining a uniform topological graph Gamma, in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over Gamma. It is established that for any cofinite graph there exists a unique cofinite completion.