Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Traces and Euler characteristics
- 2 Groups of virtually finite dimension
- 3 Free abelianised extensions of finite groups
- 4 Arithmetic groups
- 5 Topological methods in group theory
- 6 An example of a finite presented solvable group
- 7 SL3(Fq[t]) is not finitely presentable
- 8 Two-dimensional Poincaré duality groups and pairs
- 9 Metabelian quotients of finitely presented soluble groups are finitely presented
- 10 Soluble groups with coherent group rings
- 11 Cohomological aspects of 2-graphs. II
- 12 Recognizing free factors
- 13 Trees of homotopy types of ( m)-complexes
- 14 Geometric structure of surface mapping class groups
- 15 Cohomology theory of aspherical groups and of small cancellation groups
- 16 Finite groups of deficiency zero
- 17 Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen
- 18 Two-dimensional complexes with torsion values not realizable by self-equivalences
- 19 Applications of Nielsen's reduction method to the solution of combinatorial problems in group theory: a survey
- 20 Chevalley groups over polynomial rings
- List of problems
1 - Traces and Euler characteristics
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Traces and Euler characteristics
- 2 Groups of virtually finite dimension
- 3 Free abelianised extensions of finite groups
- 4 Arithmetic groups
- 5 Topological methods in group theory
- 6 An example of a finite presented solvable group
- 7 SL3(Fq[t]) is not finitely presentable
- 8 Two-dimensional Poincaré duality groups and pairs
- 9 Metabelian quotients of finitely presented soluble groups are finitely presented
- 10 Soluble groups with coherent group rings
- 11 Cohomological aspects of 2-graphs. II
- 12 Recognizing free factors
- 13 Trees of homotopy types of ( m)-complexes
- 14 Geometric structure of surface mapping class groups
- 15 Cohomology theory of aspherical groups and of small cancellation groups
- 16 Finite groups of deficiency zero
- 17 Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen
- 18 Two-dimensional complexes with torsion values not realizable by self-equivalences
- 19 Applications of Nielsen's reduction method to the solution of combinatorial problems in group theory: a survey
- 20 Chevalley groups over polynomial rings
- List of problems
Summary
Let A be a ring. I shall write p(A) for the category of finitely generated projective right A-modules and K0 (A) for its Grothendieck group. When A is an algebra over some commutative ring k let pk(A) denote the category of right A-modules M such that M ϵ P(k), the category of ‘k representations’ of A, and let Rk(A) denote its Grothendieck group.
I shall be mainly concerned with P(A) in the case when A = kG, the group algebra of a group G, and particularly the case when k = Z. This is a subject that barely exists except for some very special classes of groups G, notably finite groups and abelian groups. The following questions indicate the level of our ignorance.
Let G be a torsion free group.
(i) Is every P ϵ P(ZG) free?
No in general but there is essentially only one example known [D], Dunwoody's trefoil module. G = 〈x, y ∣x2 = y3 〉 is the trefoil group, P is a relation module arising from a presentation of G, and
P ⊕ ZG ≅ ZG ⊕ ZG.
(ii) Is K0(ZG) ≅ Z?
No counterexamples are known.
I mention in passing the following classical problem, which turns out to be related to the above questions in certain cases.
(iii) Is ZG without non trivial 0-divisors.
(iv) […]
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- Homological Group Theory , pp. 1 - 26Publisher: Cambridge University PressPrint publication year: 1979
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