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Register a new userOn growth of homology torsion in amenable 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.
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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.
The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups $G$ with the following properties: (i) $G$ is finitely generated, (ii) $G$ is amenable, e.g. of intermediate growth, (iii) any finitely presented group $E$ with a quotient isomorphic to $G$ contains non-abelian free subgroups, or the stronger (iii) any finitely presented group with a quotient isomorphic to $G$ is large.
We define a notion of uniform density on translation bounded measures in unimodular amenable locally compact Hausdorff groups, which is based on a group invariant introduced by Leptin in 1966. We show that this density notion coincides with the well-known Banach density. We use Leptin densities for a new geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.
We generalize a result of R. Thomas to establish the non-vanishing of the first l2-Betti number for a class of finitely generated groups.
We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is Z/2.
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