On homology torsion growth

Miklos Abert (1) , Nicolas Bergeron (2) , Mikolaj Fraczyk (3) , Damien Gaboriau (4)

1 Alfréd Rényi Institute of Mathematics
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 University of Chicago
4 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Mots clés: torsion in homology; torsion growth; arithmetic lattices; Artin groups; mapping class groups

Domaines: Topologie géométrique [math.GT]; Topologie algébrique [math.AT]; Théorie des groupes [math.GR]; Théorie des nombres [math.NT]

Miklos Abert, Nicolas Bergeron, Mikolaj Fraczyk, Damien Gaboriau. On homology torsion growth. Journal of the European Mathematical Society, In press, ⟨10.48550/arXiv.2106.13051⟩. ⟨hal-03270482v2⟩