{"paper":{"title":"Free Groups in Quaternion Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"A. C. Souza Filho, S. O. Juriaans","submitted_at":"2009-01-14T13:26:34Z","abstract_excerpt":"In \\cite{jpsf} we constructed pairs of units $u,v$ in $\\Z$-orders of a quaternion algebra over $\\Q (\\sqrt{-d})$, $d \\equiv 7 \\pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\\in \\mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\\ \\mathbb{Q}$, thus including matrix algebras. More precisely, we show that $< u^n,v^n> $ is a free group for all $n\\geq 1$ and $d>2$ and for $d=2$ and all $n\\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also establishe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1977","kind":"arxiv","version":5},"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"}