Ends—group Theoretical and Topological Aspects

This is a survey of topological, group theoretical and some graph theoretical aspects of ends. After discussing the notion of ends in topology, we consider ends of graphs and show that the metric end topology of connected graphs is metrizable. The “1–2–Cantor theorem” is proved for graphs whose ends are all limit ends, that is, ends which are accumulation points of an orbit of the group of automorphisms. We also discuss ends of finitely generated groups, Stallings’ Structure Theorem and further applications concerning the cycle space of a graph and random walks.