Our idea of constructing a cofinite graph starts by defining a uniform topological graph Gamma. This idea is motivated by cofinite groups structure, due to B. Hartley.
We define and establish a theory of cofinite connectedness of a cofinite graph. We found that if G is a cofinite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable cofinite uniform topological structure so that X generates G topologically if and only if Gamma is cofinitely connected.
Next we develop group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids,
thus leading to the notions of cofinite groupoids.
This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).