Infinite subgroups of the Morava stabilizer groups
V. Gorbounov
M. Mahowald
P. Symonds
We discuss certain infinite subgroups of the Morava
stabilizer groups and outline some applications
in homotopy theory.
Consider a cyclic algebra ${\Bbb D}$ over ${\Bbb Q_p}$ of
index $p-1$ with Hasse invariant ${1\over {p-1}}$.
Let ${\Bbb S}l$ be the group of strict units of ${\Bbb D}$ of
reduced norm one.
The main result is the following:
\begin{thm}\label{one} There is a p-th root of unity $\alpha\in {\Bbb
S}l$ and a (p-1)-st root of $-p$ in ${Bbb D}$ such that
\begin{enumerate}
\item \{$\displaystyle \alpha^{\omega^i}$, $1\leq i\leq p-1$\} generate
a subgroup $G$ of ${\Bbb S}l$, which is isomorphic to a free product
of $p-1$ copies of ${\Bbb Z/p}$:
${\Bbb Z/p}*\cdots *{\Bbb Z/p}$
\item $G$ is dense in ${\Bbb S}l$.
\end{enumerate}
\end{thm}