2012/11/14 by Sergei Buyalo, Buyalo, Sergei, Viktor Schroeder +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.DG #math.MG #msc:53C23 #msc:53C35
paper · pdf · doi:10.48550/arxiv.1211.3237
57 pages. arXiv admin note: substantial text overlap with arXiv:1012.1699
arxiv created 2012/11/14 · arxiv updated 2012/11/15
A Moebius structure (on a set X) is a class of metrics having the same cross-ratios. A Moebius structure is ptolemaic if it is invariant under inversion operations. The boundary at infinity of a CAT(-1) space is in a natural way a Moebius space, which is ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Moebius geometry: Let X be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then X is Moebius equivalent to the boundary at infinity of a rank one symmetric space.