On homology torsion growth
arXiv:2106.13051
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.
51 pages, 3 figures. Modifications after referee's recommandations. A section added of "1.3-speculations and questions" relating the homology growth with the sofic entropy Betti numbers. Name change: Right-angled groups become Chain-commuting groups (see Note 1, p. 8). Comments and a new reference added for the proof of Proposition 10.15 that was incomplete. To be published in J. Eur. Math. Soc