Whitehead groups of certain hyperbolic manifolds

1984 ◽  
Vol 95 (2) ◽  
pp. 299-308 ◽  
Author(s):  
A. J. Nicas ◽  
C. W. Stark
Keyword(s):  
Class Group ◽  
Universal Cover ◽  
Hyperbolic Manifolds ◽  
Fundamental Groups ◽  
Whitehead Group ◽  
Free Products ◽  
Projective Class ◽  
Aspherical Manifold ◽  
Hyperbolic Polyhedra ◽  
Whitehead Groups

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

scholarly journals Whitehead groups of semidirect products of free groups II

1980 ◽  
Vol 21 (1) ◽  
pp. 71-74 ◽  
Author(s):  
Koo-Guan Choo
Keyword(s):  
Group Ring ◽  
Class Group ◽  
Free Groups ◽  
Integral Group Ring ◽  
Integral Group ◽  
Whitehead Group ◽  
Semidirect Products ◽  
Projective Class ◽  
Whitehead Groups

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . For undefined terminologies used in the paper, we refer to [3] and [6].

Invariance of Torsion and the Borsuk Conjecture

1980 ◽  
Vol 32 (6) ◽  
pp. 1333-1341 ◽  
Author(s):  
T. A. Chapman
Keyword(s):  
Class Group ◽  
Homotopy Equivalent ◽  
Homotopy Equivalence ◽  
Good Sense ◽  
Whitehead Group ◽  
Simple Homotopy ◽  
Direct Sum ◽  
Projective Class ◽  
Arbitrary Space ◽  
Compact Polyhedron

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.

scholarly journals Whitehead groups of semidirect products of free groups

1978 ◽  
Vol 19 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Koo-Guan Choo
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Semidirect Product ◽  
Class Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Whitehead Group ◽  
Infinite Cyclic Group ◽  
Semidirect Products ◽  
Projective Class

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

scholarly journals Stably free resolutions of lattices over finite groups

1990 ◽  
Vol 49 (3) ◽  
pp. 364-385
Author(s):  
K. W. Gruenberg
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Class Group ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Residue Classes ◽  
Projective Class

AbstractFor a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

scholarly journals C*-SIMPLE GROUPS: AMALGAMATED FREE PRODUCTS, HNN EXTENSIONS, AND FUNDAMENTAL GROUPS OF 3-MANIFOLDS

2011 ◽  
Vol 03 (04) ◽  
pp. 451-489 ◽  
Author(s):  
PIERRE DE LA HARPE ◽  
JEAN-PHILIPPE PRÉAUX
Keyword(s):  
Sufficient Conditions ◽  
Simple Groups ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Groups Acting On Trees ◽  
Jsj Decompositions ◽  
Fixed Point Sets

We establish sufficient conditions for the C*-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their nontrivial subnormal subgroups; for example normal subgroups of Baumslag–Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser–Milnor and JSJ decompositions. Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree T and on its boundary ∂T, and faithfulness in a strong sense. An important step in this analysis is to identify automorphisms of T which are slender, namely such that their fixed-point sets in ∂T are nowhere dense for the shadow topology.

On the Algebra of Semigroup Diagrams

1997 ◽  
Vol 07 (03) ◽  
pp. 313-338 ◽  
Author(s):  
Vesna Kilibarda
Keyword(s):  
Homotopy Theory ◽  
Structural Information ◽  
Geometric Method ◽  
Fundamental Groups ◽  
Free Products ◽  
Semigroup Presentation ◽  
Fundamental Groupoid ◽  
Low Dimensional ◽  
Semigroup Diagrams ◽  
Natural Way

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

scholarly journals UNCOUNTABLY MANY ARCS IN S3 WHOSE COMPLEMENTS HAVE NON-ISOMORPHIC, INDECOMPOSABLE FUNDAMENTAL GROUPS

2000 ◽  
Vol 09 (04) ◽  
pp. 505-521 ◽  
Author(s):  
ROBERT MYERS
Keyword(s):  
Fundamental Group ◽  
Fundamental Groups ◽  
Free Products

An uncountable collection of arcs in S3 is constructed, each member of which is wild precisely at its endpoints, such that the fundamental groups of their complements are non-trivial, pairwise non-isomorphic, and indecomposable with respect to free products. The fundamental group of the complement of a certain Fox-Artin arc is also shown to be indecomposable.

ℓ∞-COHOMOLOGY AND METABOLICITY OF NEGATIVELY CURVED COMPLEXES

1999 ◽  
Vol 09 (01) ◽  
pp. 51-77 ◽  
Author(s):  
IGOR MINEYEV
Keyword(s):  
Simplicial Complex ◽  
Universal Cover ◽  
Fundamental Groups ◽  
Sufficient Criterion ◽  
Positive Dimension ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Higher Dimensional ◽  
Isoperimetric Functions ◽  
Finite Simplicial Complex

We prove the analog of de Rham's theorem for ℓ∞-cohomology of the universal cover of a finite simplicial complex. A sufficient criterion is given for linearity of isoperimetric functions for filling cycles of any positive dimension over ℝ. This implies the linear higher dimensional isoperimetric inequalities for the fundamental groups of finite negatively curved complexes and of closed negatively curved manifolds. Also, these groups are ℝ-metabolic.

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