The boundary of a Busemann space

Abstract

Abstract. Let X be a proper Busemann space. Then there is a well defined boundary, ∂X, for X. Moreover, if X is (Gromov) hyperbolic (resp. nonpositively curved), then this boundary is homeomorphic to the hyperbolic (resp. non-positively curved) boundary. §0. IntroductionThe boundary of a (Gromov) hyperbolic space (and hence of a (Gromov) hyperbolic group) was introduced in Gromov's now famous article on hyperbolic groups [G1]. Since then, this notion has received much attention and provided many interesting results (see [F] , [G1], [GH], [Sw]). This notion of boundary has been generalized to non-positively curved spaces and automatic groups (see [G2] and [NS]respectively) although it appears that the proof of this for non-positively curved spaces has not been published anywhere. However, the notion of boundary can also be extended to more general class of spaces called Busemann spaces which were defined in [Bo]. In this paper, we provide an elementary proof that the boundary of a Busemann space is well defined.Definition. Let X be a proper Busemann space, and let x 0 ∈ X. We define the boundary of X relative to x 0 as ∂ x0 X = {f : [0, ∞) → X | where f (0) = x 0 and f is an isometry} and we give ∂ x0 X the compact-open topology. Our main theorem is the following:Main Theorem. Let X be a proper Busemann space and let x 0 and x 1 be two distinct points in X. Then ∂ x0 X is homeomorphic to ∂ x1 X. (resp. non-positively curved), then ∂ x0 X is homeomorphic to the hyperbolic (resp. non-positively curved) boundary. Corollary. If a Busemann space X is (Gromov) hyperbolicIn §1 we give the basic definitions and some background material, and in §2 we define the boundary of a Busemann space. The author would like to thank his advisor, Edward C. Turner, for all his helpful suggestions and support.

A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann

“…Previous papers were devoted to Alexandrov non-positively curved spaces. Here we deal with Busemann spaces defined in [4], see also [5]- [7]).…”