On homology torsion growth

  • Miklos Abert

    Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Nicolas Bergeron

    ENS / PSL University, Paris, France
  • Mikołaj Frączyk

    University of Chicago, USA
  • Damien Gaboriau

    CNRS / ENS de Lyon, France
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Abstract

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Cite this article

Miklos Abert, Nicolas Bergeron, Mikołaj Frączyk, Damien Gaboriau, On homology torsion growth. J. Eur. Math. Soc. 27 (2025), no. 6, pp. 2293–2357

DOI 10.4171/JEMS/1411