Free Groups in Quaternion Algebras

TL;DR

This paper extends previous work on units in quaternion algebra orders, proving that certain generated groups are free for all positive integers n under specified conditions, using Pell's and Gauss' equations.

Contribution

It generalizes the construction of free groups generated by units in quaternion algebra orders to all imaginary quadratic extensions, including matrix algebras.

Findings

Units from Pell's and Gauss' equations generate free groups for all n≥1 when d>2.

For d=2, the free group property holds for all n≥2.

A criterion for free semigroup generation by homeomorphisms is established.

Abstract

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.