Formality properties of finitely generated groups and Lie algebras

Abstract We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

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Some cancellation theorems with applications to nilpotent groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700018152 ◽

1977 ◽

Vol 23 (2) ◽

pp. 147-165 ◽

Cited By ~ 26

Author(s):

R. Hirshon

Keyword(s):

Cyclic Group ◽

Nilpotent Groups ◽

Direct Products ◽

Normal Subgroups ◽

Finitely Generated ◽

Maximal Condition ◽

Infinite Cyclic Group ◽

Torsion Free

AbstractIf C is a group which satisfies the maximal condition for normal subgroups, then C may be cancelled from a group A in direct products if and only if the infinite cyclic group can be cancelled from A. Finitely generated torsion free nilpotent groups of class 2 satisfy a Remak Krull Schmidt condition.

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Direct decompositions of finitely generated torsion-free nilpotent groups

Mathematische Zeitschrift ◽

10.1007/bf01214492 ◽

1975 ◽

Vol 145 (1) ◽

pp. 1-10 ◽

Cited By ~ 9

Author(s):

Gilbert Baumslag

Keyword(s):

Nilpotent Groups ◽

Finitely Generated ◽

Direct Decompositions ◽

Torsion Free

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On the Residual Finiteness of Polygonal Products of Nilpotent Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-052-8 ◽

1992 ◽

Vol 35 (3) ◽

pp. 390-399 ◽

Cited By ~ 2

Author(s):

Goansu Kim ◽

C. Y. Tang

Keyword(s):

Nilpotent Groups ◽

Vertex Group ◽

Residual Finiteness ◽

Finitely Generated ◽

Residually Finite ◽

Cyclic Subgroups ◽

Isolated Subgroup ◽

Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

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Genus of nilpotent groups of Hirsch length six

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007328x ◽

1995 ◽

Vol 117 (3) ◽

pp. 431-438 ◽

Cited By ~ 1

Author(s):

Charles Cassidy ◽

Caroline Lajoie

Keyword(s):

Finite Number ◽

Nilpotent Group ◽

Nilpotent Groups ◽

Finitely Generated ◽

Torsion Free

AbstractIn this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

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The Nowicki conjecture for free metabelian Lie algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500954 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050095

Author(s):

Vesselin Drensky ◽

Şehmus Fındık

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Polynomial Algebra ◽

Vector Subspace ◽

Classical Theorem ◽

Finitely Generated ◽

Commutator Ideal ◽

Ideal Formula ◽

Locally Nilpotent ◽

Derivation Formula

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

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Group Cohomology and Lp-Cohomology of Finitely Generated Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-2003-027-x ◽

2003 ◽

Vol 46 (2) ◽

pp. 268-276 ◽

Cited By ~ 7

Author(s):

Michael J. Puls

Keyword(s):

Banach Space ◽

Cohomology Group ◽

Nilpotent Groups ◽

Group Cohomology ◽

Infinite Group ◽

Finitely Generated ◽

Finitely Generated Groups ◽

Cohomology Space ◽

First Cohomology Group

AbstractLet G be a finitely generated, infinite group, let p > 1, and let Lp(G) denote the Banach space . In this paper we will study the first cohomology group of G with coefficients in Lp(G), and the first reduced Lp-cohomology space of G. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

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Decision problems concerning S-arithmetic groups

Journal of Symbolic Logic ◽

10.2307/2274327 ◽

1985 ◽

Vol 50 (3) ◽

pp. 743-772 ◽

Cited By ~ 9

Author(s):

Fritz Grunewald ◽

Daniel Segal

Keyword(s):

Additive Group ◽

Nilpotent Groups ◽

Isomorphism Problem ◽

Algorithmic Problem ◽

Finitely Generated ◽

Finitely Presented Groups ◽

Finite Dimensional ◽

Associative Rings ◽

Finitely Presented ◽

Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

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Finitely generated nilpotent group C*-algebras have finite nuclear dimension

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0049 ◽

2018 ◽

Vol 2018 (738) ◽

pp. 281-298 ◽

Cited By ~ 1

Author(s):

Caleb Eckhardt ◽

Paul McKenney

Keyword(s):

Irreducible Representation ◽

Nilpotent Group ◽

Nilpotent Groups ◽

Irreducible Representations ◽

Finitely Generated ◽

C Algebra ◽

C Algebras ◽

K Theory ◽

Universal Coefficient Theorem ◽

Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

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