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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.
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Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $tilde X$ with compact quotient $X$. Fix $n geq 1$, assume $H_n(tilde X, mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.
Torsion sensitive intersection homology was introduced to unify severa
Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $ell^2$-homology of Sigma vanishes in all but the middle dimension.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
We calculate the singular homology and v{C}ech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Edas proof that the first singular homology groups of these spaces are isomorphic.
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