Mathematics – Group Theory
Scientific paper
2009-01-14
Mathematics
Group Theory
10 pages, article presented in conferences: Algebra School, Brasilia-Brazil, Brasilia National University (july-2010); Summer
Scientific paper
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 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.
Juriaans Stanley Orlando
Souza Filho A. C.
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