Title: A new notion of rank for finite supersolvable groups and
free linear actions on products of spheres

Authors: Laurence Barker & Erg\" un Yal\c c\i n

Contact info: Dept. of Mathematics, Bilkent
University, Ankara, 06533, Turkey.


Paper Status: Published J. Group Theory 6 (2003), 347-364.

Abstract: 
For a finite supersolvable group $G$, we define the {\it saw rank}
of $G$ to be the minimum number of sections $G_k - G_{k-1}$ of a
cyclic normal series $G_*$ such that $G_k - G_{k-1}$ owns an
element of prime order. The {\it axe rank} of $G$, studied by Ray
\cite{Ray}, is the minimum number of spheres in a product of
spheres admitting a free linear action of $G$. Extending a
question of Ray, we conjecture that the two ranks are equal. We
prove the conjecture in some special cases, including that where
the axe rank is $1$ or $2$. We also discuss some relations between
our conjecture and some questions about Bieberbach groups and free
actions on tori.