2018/10/07 by Sergei Buyalo, Buyalo, Sergei · 1 citation
Mathematics · #51B10 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:51B10
paper · pdf · doi:10.48550/arxiv.1810.03133
17 pages
arxiv created 2018/10/07 · arxiv updated 2018/10/09
Any (boundary continuous) hyperbolic space induces on the boundary at infinity a Moebius structure which reflects most essential asymptotic properties of the space. In this paper, we initiate the study of the inverse problem: describe Moebius structures which are induced by hyperbolic spaces at least in the simplest case of the circle. For a large class of Moebius structures on the circle, we define a canonical "filling" each of them, which serves as a natural candidate for a solution of the inverse problem. This is a 3-dimensional (pseudo)metric space Harm, which consists of harmonic 4-tuples of the respective Moebius structure with a distance determined by zig-zag paths. Our main result is the proof that every line in Harm is a geodesic, i.e., shortest in the zig-zag distance on each segment. This gives a good starting point to show that Harm is Gromov hyperbolic with the prescribed Moebius structure at infinity.