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

Abstract

I 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. The main results of this paper are twofold. In the first part, I focus on questions that occurred in generalizing Poincaré's Algorithm to my symmetric space. I 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, I study further questions related to hyperplanes and Dirichlet–Selberg domains in my symmetric space. I 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.