Free Groups in Quaternion Algebras

الأصل · EN

In jpsf we constructed pairs of units u,v in -orders of a quaternion algebra over (√-d), d ≡ 7 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.