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We discussed some properties of a family of symmetric spaces, namely$\mathcal{P}(n):=SL(n,\mathbb{R})/SO(n,\mathbb{R})$, where we replace theRiemannian 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 Poincar\'e's fundamental polyhedron theorem.

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