Cofinite Connectedness and Cofinite Group Actions

TL;DR

This paper develops the concept of cofinite connectedness in cofinite graphs, explores its properties, and characterizes group actions on such graphs to ensure uniform continuity, linking graph structure with group topology.

Contribution

It introduces cofinite connectedness for cofinite graphs and characterizes group actions that induce compatible cofinite topologies, bridging graph theory and topological group actions.

Findings

Cofinite connectedness parallels classical connectedness properties.

Cayley graphs of cofinite groups can be endowed with cofinite uniform structures.

Group actions on cofinite graphs can be uniformly continuous with appropriate structures.

Abstract

We have defined and established a theory of cofinite connectedness of a cofinite graph. Many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. We have seen that if $G$ is a cofinite group and Gamma=Gamma(G,X) is the Cayley graph. Then Gamma can be given a suitable cofinite uniform topological structure so that $X$ generates $G$, topologically iff Gamma is cofinitely connected. Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to 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.