On homology torsion growth

Miklos Abert, Nicolas Bergeron, Mikolaj Fraczyk, Damien Gaboriau


To appear in Journal of the European Mathematical Society

Download: pdf (1.4 Mo)
Hal: https://hal.archives-ouvertes.fr/hal-03270482v1
arXiv: https://arxiv.org/abs/2106.13051


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.