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Register a new userSome remarks on the homology of nilpotent groups
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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.
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The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_infty$-property and are not co-Hopfian for $n ge 3$.
We give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra.
For an infinite field $F$, we study the cokernel of the map of homology groups $H_{n+1}(mathrm{GL}_{n-1}(F),mathbb{k}) to H_{n+1}(mathrm{GL}_{n}(F),mathbb{k})$, where $mathbb{k}$ is a field such that $(n-2)!in mathbb{k}^times$, and the kernel of the natural map $H_{n}big(mathrm{GL}_{n-1}(F),mathbb{Z}big[frac{1}{(n-2)!} big] big) to H_{n}big(mathrm{GL}_{n}(F),mathbb{Z}big[frac{1}{(n-2)!} big]big)$. We give conjectural estimates of these cokernel and kernel and prove our conjectures for $nleq 4$.
For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $infty$-categories) for additive $infty$-categories, (2) define the notion of nilpotent extensions for suitable $infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $infty$-categories which are not monogenically generated (such as the stable $infty$-category of Voevodskys motives or the stable $infty$-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarkos notion of weight structures which provides a ring-with-many-objects analog of a connective $mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work of Hoyois-Krishna for homotopy $K$-theory, and establish new cases of Blancs lattice conjecture.
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