{"paper":{"title":"Limit sets for modules over groups on CAT(0) spaces -- from the Euclidean to the hyperbolic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Robert Bieri, Ross Geoghegan","submitted_at":"2013-06-14T14:11:01Z","abstract_excerpt":"The observation that the 0-dimensional Geometric Invariant $\\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\\'e's limit set $\\Lambda (\\Gamma)$ of a discrete group $\\Gamma $ of M\\\"obius transformations (which contains the horospherical limit set of $\\Gamma $) to the roots of tropical geometry (closely related to $\\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3403","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}