{"paper":{"title":"Generic thinness in finitely generated subgroups of $\\textrm{SL}_n(\\mathbb Z)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Elena Fuchs, Igor Rivin","submitted_at":"2015-06-04T21:45:25Z","abstract_excerpt":"We show that for any $n\\geq 2$, two elements selected uniformly at random from a \\emph{symmetrized} Euclidean ball of radius $X$ in $\\textrm{SL}_n(\\mathbb Z)$ will generate a thin free group with probability tending to $1$ as $X\\rightarrow \\infty.$ This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to $1$. On the other hand, we give an upper bound less than $1$ for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where $n>2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01735","kind":"arxiv","version":1},"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"}