Virtual rational Betti numbers of nilpotent-by-abelian groups

In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $mathrm{vb}_j(G):=sup_{Minmathcal{A}_G}dim_mathbb{Q} H_j(M,mathbb{Q})$ is finite for all $0leq jleq n$, where $mathcal{A}_G$ is the set of all finite index subgroups of $G$.

A note on whiteheads quadratic functor

For an abelian group $A$, we give a precise homological description of the kernel of the natural map $Gamma(A) to Aotimes_mathbb{Z} A$, $gamma(a)mapsto aotimes a$, where $Gamma$ is whiteheads quadratic functor from the category of abelian groups to itself.

Third homology of perfect central extensions

For a central perfect extension of groups $A rightarrowtail G twoheadrightarrow Q$, we study the maps $H_3(A,mathbb{Z}) to H_3(G, mathbb{Z})$ and $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$ provided that $Asubseteq G$. First we show that the image of $H_3(A, mathbb{Z})to H_3(G, mathbb{Z})/rho_ast(Aotimes_mathbb{Z} H_2(G, mathbb{Z}))$ is $2$-torsion where $rho: A times G to G$ is the usual product map. When $BQ^+$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$.

The homology of $mathrm{SL}_2$ of discrete valuation rings

Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(mathrm{SL}_2(A),mathbb{Z}[frac{1}{2}]) to H_3(mathrm{SL}_2(F),mathbb{Z}[frac{1}{2}])to mathcal{RP}_1(k)[frac{1}{2}]to 0$, where $mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $Gamma_0(mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $mathfrak{m}_A$. We also prove that there is an exact sequence $0to overline{mathcal{P}}(k)[frac{1}{2}]to H_2(Gamma_0(mathfrak{m}_A),mathbb{Z}[frac{1}{2}])to H_2(mathrm{SL}_2(A),mathbb{Z}[frac{1}{2}])to I^2(k)[frac{1}{2}]to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $mathrm{GW}(k)$ and $overline{mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $mathcal{P}(k)$ of $k$.

Some remarks on the homology of nilpotent groups

In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved.