Geometry of Selberg's bisectors in the symmetric space $SL(n,\mathbb{R})\mathrm{/}SO(n,\mathbb{R})$

We discussed some properties of a family of symmetric spaces, namely $\mathcal{P}(n):=SL(n,\mathbb{R})\mathrm{/}SO(n,\mathbb{R})$, where we replace the Riemannian metric on $\mathcal{P}(n)$ with a premetric suggested by Selberg. These properties are generalizations of properties on the hyperbolic spaces ${\mathbf{H}}^n$ related to Poincaré’s fundamental polyhedron theorem.