The paint group Tits Satake theory of hyperbolic symmetric spaces: the distance function, paint invariants and discrete subgroups

The present paper, which is partially a review, but also contains several completely new results, aims at presenting, in a unified mathematical framework, a complex and articulated lore regarding non-compact symmetric spaces, with negative curvature, whose isometry group is a non-compact, real simple Lie group. All such manifolds are Riemannian normal manifolds, according to Alekseevsky's definition, in the sense that they are metrically equivalent to a solvable Lie group manifold. This identification provides a vision in which, on one side one can derive quite explicit and challenging formulae for the unique distance function between points of the manifold, on the other one, one can organize the entire set of the available manifolds in universality classes distinguished by their common Tits Satake submanifold and, correspondingly, by their non-compact rank. The members of the class are distinguished by their different Paint Groups, the latter notion having been introduced by two of the present authors in an earlier collaboration. In relation to the construction of neural networks, these mathematical structures offer unique possibilities of replacing ad hoc activation functions with the naturally defined non-linear operations that relate Lie algebras to Lie Groups and vice-versa. The Paint Group invariants offer new tokens both to construct algorithms and inspect (hopefully to control) their working. A conspicuous part of the paper is devoted to the study and systematic construction of parabolic/elliptic discrete subgroups of the Lie groups SO(r,r+q), in view of discretization and/or tessellations of the space to which data are to be mapped. Furthermore, it is shown how the ingredients of Special K\"ahler Geometry and the c-map, well known in the supergravity literature, provide a unified classification scheme of the relevant Tits Satake universality classes with non-compact rank r<5.