A representation theorem for end spaces of infinite graphs

TL;DR

This paper establishes a topological representation for end-spaces of infinite graphs using special order trees, bridging graph theory, topology, and geometric group theory.

Contribution

It introduces a structure theorem that characterizes the end-space of any infinite graph via an order-tree representation, providing a new topological perspective.

Findings

End-spaces can be represented by special order trees.

A structure theorem links graph ends to order-tree chains.

The approach unifies concepts across multiple mathematical disciplines.

Abstract

End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology. Our main result is that every end-space can topologically be represented by a special order tree. Our main proof ingredient is a structure theorem that we introduce, which carves out the order-tree-like structure of any graph in such a way that there is a natural bijection between the ends of the graph and the limit-type down-closed chains of the order-tree.