scholarly journals Finitely Generated Algebras Associated with Rational Vector Fields

1997 ◽  
Vol 198 (2) ◽  
pp. 481-498
Author(s):  
Hisayo Aoki ◽  
Masayoshi Miyanishi
Keyword(s):  
Vector Fields ◽  
Finitely Generated ◽  
Rational Vector ◽  
Rational Vector Fields

scholarly journals Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds

1999 ◽  
Vol 59 (2) ◽  
pp. 271-295
Author(s):  
Victor Brunsden
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Group Cohomology ◽  
Finitely Generated ◽  
Rigidity Result ◽  
Finitely Generated Group ◽  
Local Rigidity ◽  
Banach Manifolds ◽  
The Stability ◽  
Smooth Closed Manifold

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

Relation modules of infinite groups, II

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Martin Evans
Keyword(s):  
Free Group ◽  
Vector Fields ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Continuous Vector ◽  
Finitely Generated Group ◽  
Number Of Generators ◽  
Free Modules

AbstractLet F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let $$\bar R$$ and $$\bar S$$ denote the associated relation modules of G. It is well known that $$\bar R \oplus (\mathbb{Z}G)^{d(G)} \cong \bar S \oplus (\mathbb{Z}G)^{d(G)}$$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations as above such that . Our approach depends on the existence of nonfree stably free modules over certain commutative rings and, in particular, on the existence of certain Hurwitz-Radon systems of matrices with integer entries discovered by Geramita and Pullman. This approach was motivated by results of Adams concerning the number of orthonormal (continuous) vector fields on spheres.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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