2009/01/14 by S. O. Juriaans, Juriaans, S. O., A. C. Souza Filho +1
Mathematics · #16S34 #16U60 #20E05 #20F67 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0901.1977
openalex publication_date 2009/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In \citejpsf we constructed pairs of units u,v in \Z-orders of a quaternion algebra over \Q (√(-d)), d ≡ 7 \pmod 8 positive and square free, such that < u^ n,vn> is free for some n∈ ℕ. Here we extend this result to any imaginary quadratic extension of ℚ, thus including matrix algebras. More precisely, we show that < un,vn> is a free group for all n≥ 1 and d>2 and for d=2 and all n≥ 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 established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free.