On the radical of group rings

Abstract It is conjectured that, for any group G , the Jacobson radical $$J({\mathbb {Z}}[G])$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the integral group ring $${\mathbb {Z}}[G]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is zero. This is known to be true when G is finite. Here we show it is true for a reasonably large class of infinite groups, including finitely generated linear groups and groups which satisfy Higman’s ‘two unique products’ condition.