Hereditarily just-infinite torsion groups with positive first $$\ell ^2$$-Betti number

Abstract We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first $$\ell ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.