四元数代数中的自由群
Free Groups in Quaternion Algebras
摘要：
在\cite{jpsf}中，我们构造了$\Q (\sqrt{-d})$上四元数代数的$\Z$ -阶中的单位对$u,v$，$d \equiv 7 \pmod 8$为正且无平方，使得$< u^ n,v^n>$对某些$n\in \mathbb{N}$是自由的.在这里，我们将这个结果推广到$\ \mathbb{Q}$的任何虚二次扩张，从而包括矩阵代数。更精确地说，我们证明了$< u^n,v^n> $对所有$n\geq 1$和$d>2$以及$d=2$和所有$n\geq 2$都是自由群。我们使用的单位来自佩尔和高斯方程。本文还建立了一个判定同胚对生成自由半群的准则，并证明了两个确定的单位生成一个自由半群，但在这种情况下，乒乓引理不能用来证明它们生成的群是自由的。
英文摘要：
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.
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