Geometry of Selberg’s bisectors in the symmetric space SL(n,R)/SO(n,R)

We discussed some properties of a family of symmetric spaces, namely SL(n,R)/SO(n,R) =: P(n), where we replace the Riemannian metric on P(n) with a premetric suggested by Selberg. These include intersecting criteria of Selberg's bisectors, the shape of Dirichlet-Selberg domains, and angle-like functions between Selberg's bisectors. These properties are generalizations of properties on the hyperbolic spaces H^n related to Poincaré's fundamental polyhedron theorem.