Book contents
- Frontmatter
- Contents
- Foreword
- Participants
- 1 Lie Methods in Growth of Groups and Groups of Finite Width
- 2 Translation numbers of groups acting on quasiconvex spaces
- 3 On a term rewriting system controlled by sequences of integers
- 4 On certain finite generalized tetrahedron groups
- 5 Efficient computation in word-hyperbolic groups
- 6 Constructing hyperbolic manifolds
- 7 Computing in groups with exponent six
- 8 Rewriting as a special case of non-commutative Gröbner basis theory
- 9 Detecting 3-manifold presentations
- 10 In search of a word with special combinatorial properties
- 11 Cancellation diagrams with non-positive curvature
- 12 Some Applications of Prefix-Rewriting in Monoids, Groups, and Rings
- 13 Verallgemeinerte Biasinvarianten und ihre Berechnung
- 14 On groups which act freely and properly on finite dimensional homotopy spheres
- 15 On Confinal Dynamics of Rooted Tree Automorphisms
- 16 An asymptotic invariant of surface groups
- 17 A cutpoint tree for a continuum
- 18 Generalised triangle groups of type (2, m, 2)
14 - On groups which act freely and properly on finite dimensional homotopy spheres
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Foreword
- Participants
- 1 Lie Methods in Growth of Groups and Groups of Finite Width
- 2 Translation numbers of groups acting on quasiconvex spaces
- 3 On a term rewriting system controlled by sequences of integers
- 4 On certain finite generalized tetrahedron groups
- 5 Efficient computation in word-hyperbolic groups
- 6 Constructing hyperbolic manifolds
- 7 Computing in groups with exponent six
- 8 Rewriting as a special case of non-commutative Gröbner basis theory
- 9 Detecting 3-manifold presentations
- 10 In search of a word with special combinatorial properties
- 11 Cancellation diagrams with non-positive curvature
- 12 Some Applications of Prefix-Rewriting in Monoids, Groups, and Rings
- 13 Verallgemeinerte Biasinvarianten und ihre Berechnung
- 14 On groups which act freely and properly on finite dimensional homotopy spheres
- 15 On Confinal Dynamics of Rooted Tree Automorphisms
- 16 An asymptotic invariant of surface groups
- 17 A cutpoint tree for a continuum
- 18 Generalised triangle groups of type (2, m, 2)
Summary
INTRODUCTION
In C. T. C. Wall conjectured that if a countable group G of finite virtual cohomological dimension, vcd G < ∞, has periodic Farrell cohomology then G acts freely and properly on ℝn × Sm for some n and m. Obviously, if a group G acts freely and properly on some ℝn × Sm then G is countable since ℝn × Sm is a separable metric space. The Farrell cohomology generalizes the Tate cohomology theory for finite groups to the class of groups G with vcd G < ∞ (see for instance Chapter X of). Wall's conjecture was proved by Johnson in some cases and Connolly and Prassidis in general.
In Prassidis showed that there are groups of infinite vcd which act freely and properly on some ℝn × Sm. In particular, it follows from results of Prassidis and Talelli that if a countable group G has periodic cohomology after 1-step then G acts freely and properly on some ℝn × Sm.
A group G is said to have periodic cohomology after κ-steps if there is a positive integer q such that the functors Hi(G;) and Hi+q(G;) are naturally equivalent for all i > κ (cf.).
- Type
- Chapter
- Information
- Computational and Geometric Aspects of Modern Algebra , pp. 208 - 228Publisher: Cambridge University PressPrint publication year: 2000
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