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

TL;DR

This paper explores the geometric properties of Selberg's bisectors within the symmetric space formed by $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, generalizing hyperbolic space concepts using a Selberg-inspired premetric.

Contribution

It introduces and analyzes properties of a premetric on $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, extending hyperbolic space results to higher-dimensional symmetric spaces.

Findings

Generalized properties of Selberg's bisectors in symmetric spaces

Extended Poincaré's fundamental polyhedron theorem to new settings

Identified geometric structures related to Selberg's premetric

Abstract

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 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 Poincar\'e's fundamental polyhedron theorem.