Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one

1996 ◽  
Vol 125 (2) ◽  
pp. 307-365 ◽  
Author(s):  
Claus Scheiderer
Keyword(s):  
Homogeneous Spaces ◽  
Cohomological Dimension ◽  
Approximation Theorems ◽  
Dimension One ◽  
Virtual Cohomological Dimension

scholarly journals Free and Properly Discontinuous Actions of Groups on Homotopy 2n-spheres

2018 ◽  
Vol 61 (2) ◽  
pp. 305-327
Author(s):  
Marek Golasiński ◽  
Daciberg Lima Gonçalves ◽  
Rolando Jimenez
Keyword(s):  
Free Group ◽  
Cyclic Group ◽  
Cohomological Dimension ◽  
Direct Products ◽  
Orbit Spaces ◽  
Homotopy Types ◽  
Finite Dimensional ◽  
The Family ◽  
Dimension One ◽  
Virtual Cohomological Dimension

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

scholarly journals Presentability by products for some classes of groups

2017 ◽  
Vol 09 (01) ◽  
pp. 27-49
Author(s):  
P. de la Harpe ◽  
D. Kotschick
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Groups ◽  
Dense Subgroups ◽  
Finite Index Subgroups ◽  
Virtual Cohomological Dimension ◽  
By Products

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

2010 ◽  
Vol 21 (06) ◽  
pp. 727-736
Author(s):  
JANG HYUN JO
Keyword(s):  
Finite Group ◽  
Algebraic Property ◽  
Cohomological Dimension ◽  
Finite Dimensional ◽  
Free Actions ◽  
Virtual Cohomological Dimension ◽  
Algebraic Counterpart

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

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