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

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arxiv 2401.04542 v2 pith:IXQR4ZQ7 submitted 2024-01-09 math.GR

classification math.GR
keywords groupsfinitefirsttorsionhereditarilyjust-infinitenormalpositive
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Virtual First Betti Number of GGS Groups

    math.GR 2025-05 accept novelty 6.0 of 10

    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.

  2. Infinite groups from the profinite point of view

    math.GR 2025-06 accept novelty 1.0 of 10

    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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