Between ends and fibers

Abstract

Let be an infinite, locally finite, connected graph with distance function δ. Given a ray P in and a constant C ≥ 1, a vertex-sequence {x n } ∞ n=0 ⊆ VP is said to be regulated by C if, for all n ∈ N, x n+1 never precedes x n on P, each vertex of P appears at most C times in the sequence, and δ P (x n , x n+1 ) ≤ C. R. Halin (Math. Ann., 157, 1964, 125-137) defined two rays to be end-equivalent if they are joined by infinitely many pairwisedisjoint paths; the resulting equivalence classes are called ends. More recently H.A.Jung (Graph Structure Theory, Contemporary Mathematics, 147, 1993, 477-484) defined rays P and Q to be b-equivalent if there exist sequences {x n } ∞ n=0 ⊆ VP and {y n } ∞ n=0 ⊆ VQ regulated by some constant C ≥ 1 such that δ(x n , y n ) ≤ C for all n ∈ N; he named the resulting equivalence classes b-fibers. Let F 0 denote the set of nondecreasing functions from N into the set of positive real numbers. The relation P ∼ f Q (called f-equivalence) generalizes Jung's condition to δ(x n , y n ) ≤ Cf(n). As f runs through F 0 , uncountably many equivalence relations are produced on the Journal of Graph Theory © 2006 Wiley Periodicals, Inc. 125126 JOURNAL OF GRAPH THEORY set of rays that are no finer than b-equivalence while, under specified conditions, are no coarser than end-equivalence. Indeed, for every there exists an "end-defining function" f ∈ F 0 that is unbounded and sublinear and such that P ∼ f Q implies that P and Q are end-equivalent. Say P ≈ Q if there exists a sublinear function f ∈ F 0 such that P ∼ f Q. The equivalence classes with respect to ≈ are called bundles. We pursue the notion of "initially metric" rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end-equivalent. In the case that contains translatable rays we give some sufficient conditions for every f-equivalence class to contain uncountably many g-equivalence classes (where lim n→∞ g(n)/f(n) = 0). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost-transitive planar map are never bundleequivalent.

Linear and projective boundaries in HNN-extensions and distortion phenomena

“…Let k; L 1. Let v be a word of length less than L 2 k 1 in G representing an element a n for some n 2 Z such that 1 .v/ does not contain the letter ak . Then,…”

“…The word 1 .v/ does not contain a letter a1 . Therefore v D a˛for some j˛j < L. Obviously, n D˛and we are done.…”

“…: By (6.14) and by the definition of v 0 , we know that`.v 0 / Ä 2 .3 2 k 5/, and moreover, since 1 .v 0 / does not contain any letter ak , for k > 2 we obtain that…”

“…It is related to a concept due to Bonnington, Richter and Watkins [1]. This concept is rather general and for instance, Tits' boundary of a CAT.0/ space (see [2,Section 9]) fits D .V; E/ D Cay.G; X / is the graph with vertex set V D G and edge set E D ¹¹g; hº W g 1 h 2 X º.…”

Linear and Projective Boundary of Nilpotent Groups

“…The basic idea behind fibers is to consider points at infinity as equivalence classes of rays (infinite paths) which stay at bounded distance "up to linear reparametrization". In 2005 Bonnington, Richter and Watkins [BRW07] modified this concept by considering rays as equivalent whenever they stay at sublinear distance "up to linear reparametrization". They were able to use this concept to prove some nice results on infinite planar graphs, but the boundary, whose elements have been called "bundles", was not topologized and not considered for groups or vertex-transitive graphs.…”