Fixed elements of infinite trees

✍ Scribed by Norbert Polat; Gert Sabidussi

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 380 KB
• Volume: 130
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)00526-w

✦ Synopsis

It is shown that any infinite tree not containing a ray has a fixed vertex or a fixed edge. The same also holds for trees with rays (not containing a subdivision of the dyadic tree) provided there are at least three ends of maximal order.

I have found it a profitable exercise of the imagination, from a philosophical point of view, to build up the conception of an injnite arborescence and to dwell on the relations of time and causality which such a concept embodies.. . So the largest idea of an arborescence is that of an infinite number of nodes with an infinite number of branches proceeding from each of them. J.J. Sylvester [4]

1. Introduction, preliminaries

Any finite tree T has a fixed element, i.e., a vertex or an edge which is invariant under any automorphism of T. For infinite trees this need no longer be the case, the simplest counterexample being the 2-way infinite path. To what infinite trees can the statement be extended?

The proof of the finite case makes use of some notion of eccentricity or centrality, usually defined in terms of the distance function of T. One shows that there is either