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. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), 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. [...]

       Hyperbolic VorDiag http://cs.smith.edu/%7Eorourke/MathOverflow/HyperbolicVorDiag.jpg

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. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), 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. [...]

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. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), 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. [...]

 textareaHyperbolic VorDiag http://cs.smith.edu/%7Eorourke/MathOverflow/HyperbolicVorDiag.jpgtextarea

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. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), 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. [...]

        Hyperbolic VorDiag http://cs.smith.edu/%7Eorourke/MathOverflow/HyperbolicVorDiag.jpg