  Let $G$ be a finite group, $u$ a Bass unit based on an element $a$ of $G$ of
prime order, and assume that $u$ has infinite order modulo the center of the
units of the integral group ring $\Z G$. It was recently proved that if $G$ is
solvable then there is a Bass unit or a bicyclic unit $v$ and a positive
integer $n$ such that the group generated by $u^n$ and $v^n$ is a non-abelian
free group. It has been conjectured that this holds for arbitrary groups $G$.
To prove this conjecture it is enough to do it under the assumption that $G$ is
simple and $a$ is a dihedral $p$-critical element in $G$. We first classify the
simple groups with a dihedral $p$-critical element. They are all of the form
$\PSL(2,q)$. We prove the conjecture for $p=5$; for $p>5$ and $q$ even; and for
$p>5$ and $q+1=2p$. We also provide a sufficient condition for the conjecture
to hold for $p>5$ and $q$ odd. With the help of computers we have verified the
sufficient condition for all $q<10000$.
