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We present a new method to construct finitely generated, residually finite,infinite torsion groups. In contrast to known constructions, a profiniteperspective enables us to control finite quotients and normal subgroups ofthese torsion groups. As an application, we describe the first examples ofresidually finite, hereditarily just-infinite groups with positive first$\ell^2$-Betti-number. In addition, we show that these groups have polynomialnormal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

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