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

Abstract

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 centre of the units of the integral group ring ZG. 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 un and vn 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.