Bernoulli shifts on additive categories and algebraic K-theory of wreath products

Rigidity results for wreath product II 1 factors

Quadratic and hermitian forms in additive and abelian categories

We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.

We prove the K- and the $L$-theoretic Farrell-Jones conjecture with coefficients in additive categories and with finite wreath products for arbitrary lattices in virtually connected Lie groups.

On the maximal quotient ring of regular group rings

Coefficients for the Farrell–Jones Conjecture

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance, the negative K-groups and all twisted Nil-groups vanish for an additive category, if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.

Let G be a multiplicative group, K a commutative ring with unit, and K(G) the group ring of G with respect to K. We say that K(G) is regular if given an x in K(G), there is a y in K(G) such that xyx = x. Using a homological characterization of regular rings which was found independently by M. Harada [2, Theorem 5] and the author, we prove that if G is locally finite, then K(G) is regular if and only if K is regular and is uniquely divisible by the order of each element in G. More generally we show that if K(G) is regular, then G is a torsion group and K is a regular ring which is uniquely divisible by the order of each element in G. A nonhomological proof of these results has been given by J. McLaughlin (unpublished). In conclusion, we show that if G is a commutative group and K is a field of characteristic not dividing the order of any element in G, then the weak global dimension of K(G) equals the rank of G. For the most part we follow the conventions and notations in [']. Let R be a ring (with unit) and A a left R-module. The weak left dimension of A is defined as follows (see [1, Chapter VI, Exercise 3]):

Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We generalize the result presented in the book of J. Meldrum [11] also the results of A. Woryna [4]. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. We strengthen the results from the author [17, 19] and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. We generalise the results of Meldrum J. [11] about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group A does not have to act faithfully). The commutator of such a group, its minimal generating set and the center of such products has been investigated here. The minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found.

Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not faithfully). Also commutator of such group and its minimal generating set. Also center of such products was investigated. Also fundamental group of orbits of a Morse function $f:M\to \mathbb{R}$ defined on a Mebius band $M$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$ is investigated by us. The paper describes precise algebraic structure of the group $\pi_1 O(f)$. A minimal set of generators for the group of orbits of functions ${{\pi }_{1}}({{O}_{f}},f)$ arising under the action of diffeomorphisms group stabilizing the function $f$ and stabilizing $\partial M$ is found. The the Morse function $f$ has critical sets with one saddle point. The quotient group of restricted wreath products by its commutator was found. The generic sets of commutator of wreath product were investigated. Minimal generating set for this group and for commutator of group are found. This paper after previous Arxiv versions from 2019 \cite{SkArM, SkArM3} with previous title "Minimal generating set and structure of wreath product of groups with non-faithful action, comutator subgroup of wreath product and the fundamental group of orbit of Morse function $\pi_1 O(f)$" was published \cite{SkRendi}.

On permutation characters of wreath products

Unique decomposition and isomorphic refinement theorems in additive categories

Wreath product, a powerful construction in group theory, has found extensive applications in various areas of mathematics and computer science. In this paper, we present a comprehensive analysis of coding matrices associated with wreath products. The coding matrices for the wreath product of two cyclic finite groups were given for the first time. It gives a generalization of the coding matrices for the semi-direct product. We found out that the coding matrix of wreath product really has the same shape as the one for semidirect product and gave the RW-matrix for the coding matrix. An example was showed to illustrate the assertions. Conditions were also given for different wreath products of cyclic groups and that gives different orders for the wreath products.

For a large class of C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras A, we calculate the K-theory of reduced crossed products A⊗G⋊rG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\otimes G}\\rtimes _rG$$\\end{document} of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras of wreath products H≀G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\wr G$$\\end{document} for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

Conditions are given under which the self-injectivity of the group ring AG implies the finiteness of G. It has been known for some time that if A is a self-injective ring (associative with 1) and G is a finite group, then the group ring AG is selfinjective; and conversely that if AG is self-injective then A is self-injective and G is locally finite [4]. Whether or not G must actually be finite, has been studied by Gentile [2] who obtained an affirmative answer in case A is a commutative ring which is torsion-free as a Z-module. His result is included in the following theorem (see Corollary). We denote the Jacobson radical of a ring A by Rad A, and use o(H) to denote the order of a group H. THEOREM. If AG is a seif-injective group ring and o(H) is a unit in A/Rad A for allfinite subgroups H of G, then (A is seif-injective and) G is finite. PROOF. Since AG is self-injective, AG/Rad(AG) is self-injective and (Von Neumann) regular [5] and similarly since AG self-injective=-A selfinjective, it follows that A/Rad A is regular. Since G is locally finite, Rad A = Rad(AG) nA [1] so (Rad A)G c Rad(AG) and therefore AG/Rad (AG) AG/(Rad A)G (A/Rad A)G Rad(AG)/(Rad A)G Rad((A/Rad A)G)' Since o(H) is a unit in the regular ring A/Rad A for all finite subgroups H of G, it follows that (A/Rad A)G is regular [1], and therefore has zero radical. Thus AG/Rad(AG)-(A/Rad A)G and it suffices to consider a regular self-injective group ring AG. It is easily shown that when AG is self-injective, so is AH for all subgroups H of G (see e.g. [2]), so without loss of generality we assume G is countable. Then the fundamental ideal A of AG is countably generated, and since AG is regular, a result of Kaplansky [3] shows that A is Received by the editors April 26, 1971. AMS 1970 subject classifications. Primary 16A26, 16A52; Secondary 16A30.