Free Groups in Quaternion Algebras

In \cite{jpsf} we constructed pairs of units $u,v$ in $\mathbb{Z}$-orders of a quaternion algebra over $\text{Q}(\sqrt{-d})$, $d\equiv 7(\mathrm{m}\mathrm{o}\mathrm{d}8)$ 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 $n\ge 1$ and $d>2$ and for $d=2$ and all $n\ge 2$. 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.