oneended

Oneended is a term used in mathematics, often hyphenated as one-ended, to describe certain infinite structures that have a single way to go to infinity. In graph theory, a connected infinite graph is one-ended if, after removing any finite set of vertices, the remaining graph has exactly one infinite component. Informally, there is only one “direction” toward infinity.

In geometric group theory, the term extends to finitely generated groups via their Cayley graphs. The number of ends of a group is an invariant: finite groups have zero ends, infinite cyclic groups have two ends, and more complex groups can have one end or infinitely many ends. Examples include the infinite grid Z^2 and many surface groups, which are one-ended, while the free group F_n (n ≥ 2) has infinitely many ends.

A central result related to ends of groups is Stallings’ theorem: if a finitely generated group has more than one end, it splits nontrivially as a free product with amalgamation or as an HNN extension over a finite subgroup. This reveals a close link between end structure and algebraic decomposition. Conversely, one-ended groups do not admit such a nontrivial decomposition over finite subgroups.

Typical examples help illustrate the spectrum. Finite groups have zero ends; Z and the infinite dihedral group are two-ended; Z^2 and most closed surface groups are one-ended; the free group F_n with n ≥ 2 has infinitely many ends.

The concept of one-endedness provides a framework for understanding how a structure behaves at infinity and connects combinatorial, geometric, and algebraic perspectives.