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Hereditarily just-infinite torsion groups with positive first $\ell^2$-Betti number
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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 $\ell^2$-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.
Forward citations
Cited by 2 Pith papers
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Virtual First Betti Number of GGS Groups
The derived subgroups of certain GGS groups are torsion-free, finitely generated, residually finite, and not virtually diffuse, answering a question of Kionke and Raimbault.
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Infinite groups from the profinite point of view
This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.
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