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Let $G$ be a finite group, $u$ a Bass unit based on an element $a$ of $G$ ofprime order, and assume that $u$ has infinite order modulo the center of theunits of the integral group ring $\Z G$. It was recently proved that if $G$ issolvable then there is a Bass unit or a bicyclic unit $v$ and a positiveinteger $n$ such that the group generated by $u^n$ and $v^n$ is a non-abelianfree 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 $G$ issimple and $a$ is a dihedral $p$-critical element in $G$. We first classify thesimple 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 conjectureto hold for $p>5$ and $q$ odd. With the help of computers we have verified thesufficient condition for all $q<10000$.

Bass units as free factors in integral group rings of simple groups