On homology torsion growth

TL;DR

This paper establishes new vanishing results and asymptotic bounds for the growth of higher torsion homologies in arithmetic lattices, Artin groups, and mapping class groups using a novel homotopical method.

Contribution

It introduces the effective rebuilding method, a quantitative homotopical technique for constructing small classifying spaces with controlled complexity.

Findings

Vanishing results for torsion homology growth in specific groups

Asymptotic bounds for torsion growth in principal congruence subgroups

Effective rebuilding method applicable to a wide class of residually finite groups

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.