{"paper":{"title":"Quantitative properties of convex representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GR","authors_text":"Andr\\'es Sambarino","submitted_at":"2011-04-25T10:58:46Z","abstract_excerpt":"Let $\\Gamma$ be a discrete subgroup of $\\textrm{PGL}(d,\\R)$ and fix some euclidean norm $\\|\\ \\|$ on $\\R^d.$ Let $N_\\Gamma(t)$ be the number of elements in $\\Gamma$ whose operator norm is $\\leq t.$ In this article we prove an asymptotic for the growth of $N_\\Gamma(t)$ when $t\\to\\infty$ for a class of $\\Gamma$'s which contains, in particular, Hitchin representations of surface groups and groups dividing a convex set of $\\P(\\R^d).$ We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4705","kind":"arxiv","version":2},"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"}