Real valued functions and metric spaces quasi-isometric to trees

Abstract

Abstract. We prove that if X is a complete geodesic metric space with uniformly generated first homology group and f : X → R is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space.

Bushy pseudocharacters and group actions on quasitrees

“…Then scale f so that f .G/ misses Z C The edges correspond to components of f 1 .Z C 1 2 /, each of which is some possibly infinite track which separates z K into two components. This construction is also the starting point in the author's [14] where given a real valued function on a geodesic space we give a sufficient condition for the space to be quasi-isometric to a tree.…”

Bushy pseudocharacters and group actions on quasi-trees

“…The edges correspond to components of f −1 (Z + 1 2 ), each of which is some possibly infinite track which separates K into two components. This construction is also the starting point in [14] where given a real valued function on a geodesic space we give a sufficent condition for the space to be quasi-isometric to a tree.…”

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

“…This characterization has proved to be very useful, see for example [6]. For some other relations with (BP) see [15,31] and the references therein.…”