{"paper":{"title":"The paint group Tits Satake theory of hyperbolic symmetric spaces: the distance function, paint invariants and discrete subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DG","authors_text":"Mario Trigiante, Pietro Fr\\'e, Ugo Bruzzo","submitted_at":"2025-02-28T16:39:33Z","abstract_excerpt":"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 formula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.07626","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.07626/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}