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

We study several problems about the symmetric space associated with the Lie group SL(n, R). These problems are connected to an algorithm based on Poincaré’s Fundamental Polyhedron Theorem, designed to determine generalized geometric finiteness properties for subgroups of SL(n, R). The algorithm is analogous to the original one in hyperbolic spaces, while the Riemannian distance is replaced by an SL(n, R)-invariant premetric.\n The main results of this article are twofold. In the first part, we focus on questions that occurred in generalizing Poincaré’s algorithm to our symmetric space. We describe and implement an algorithm that computes the face-poset structure of finitely-sided polyhedra, and construct an angle-like function between hyperplanes. In the second part, we study further questions related to hyperplanes and Dirichlet-Selberg domains in our symmetric space. We establish several criteria for the disjointness of hyperplanes and classify particular Abelian subgroups of SL(3, R) based on whether their Dirichlet-Selberg domains are finitely-sided or not.