A representation theorem for end spaces of infinite graphs

Abstract

End spaces of infinite graphs sit at the interface between graph theory, group theory and topology. They arise as the boundary of an infinite graph in a standard sense generalising the theory of the Freudenthal boundary developed by Freudenthal and Hopf in the 1940's for infinite groups.A long-standing quest in infinite graph theory with a rich body of literature seeks to describe the possible end structures of graphs by a set of low-complexity representatives. In this paper we present a solution to this fifty-year-old problem by showing that every end space is homeomorphic to the end space of some (uniform graph on a) special order tree.