Although I cannot address your specific question, of whether bisectors "play an important role in the study of higher rank symmetric spaces," I can say that bisectors are the essence of [Voronoi diagrams][1]. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" ([arXiv link)][2], could well be relevant: > We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that *bisectors* in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. [...] <br /> ![Hyperbolic VorDiag][3] [1]: http://en.wikipedia.org/wiki/Voronoi_diagram [2]: http://arxiv.org/abs/0903.3287 [3]: http://cs.smith.edu/~orourke/MathOverflow/HyperbolicVorDiag.jpg