scholarly journals Non-Left-Orderable 3-Manifold Groups

2005 ◽  
Vol 48 (1) ◽  
pp. 32-40 ◽  
Author(s):  
Mieczysław K. Dąbkowski ◽  
Józef H. Przytycki ◽  
Amir A. Togha
Keyword(s):  
Special Interest ◽  
Branched Covers ◽  
Branched Coverings ◽  
Figure Eight Knot ◽  
Torsion Free

AbstractWe show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Homology of Branched Coverings of 3-Manifolds

1992 ◽  
Vol 44 (1) ◽  
pp. 119-134
Author(s):  
John Hempel
Keyword(s):  
Classical Result ◽  
Quantitative Information ◽  
Cyclic Cover ◽  
Branched Covers ◽  
Branched Coverings ◽  
Homology Groups

AbstractWe give a relation between the homology groups H1() and H1 (M) for a branched cyclic cover → M of arbitrary closed, oriented 3-manifolds which generalizes a classical result of Plans on covers of S3 branched over a knot and provides other quantitative information as well. We include a general "free calculus" procedure for computing homology groups of branched covers and reinterpret the results in this computational setting.

scholarly journals On the homology of branched coverings of 3-manifolds

2014 ◽  
Vol 213 ◽  
pp. 21-39 ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Topological Invariant ◽  
Ideal Class ◽  
Class Groups ◽  
Branched Covers ◽  
Branched Coverings ◽  
3 Dimensional ◽  
Genus Theory ◽  
Ideal Class Groups ◽  
Galois Covers ◽  
Insight Into

AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

scholarly journals On the homology of branched coverings of 3-manifolds

2014 ◽  
Vol 213 ◽  
pp. 21-39 ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Topological Invariant ◽  
Ideal Class ◽  
Class Groups ◽  
Branched Covers ◽  
Branched Coverings ◽  
3 Dimensional ◽  
Genus Theory ◽  
Ideal Class Groups ◽  
Galois Covers ◽  
Insight Into

AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

scholarly journals The isomorphism problem for a class of para-free groups

1997 ◽  
Vol 40 (3) ◽  
pp. 541-549 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger ◽  
Michael Stille
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Special Interest ◽  
Isomorphism Problem ◽  
Combinatorial Group Theory ◽  
History Of ◽  
One Relator Groups ◽  
Rank 2 ◽  
Combinatorial Group ◽  
Torsion Free

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

scholarly journals THE FIGURE EIGHT KNOT GROUP IS CONJUGACY SEPARABLE

2009 ◽  
Vol 08 (04) ◽  
pp. 539-556 ◽  
Author(s):  
S. C. CHAGAS ◽  
P. A. ZALESSKII
Keyword(s):  
Figure Eight Knot ◽  
Free Subgroups ◽  
Conjugacy Separable ◽  
Torsion Free

We prove that torsion free subgroups of GL2(ℂ) (abstractly) commensurable with the Euclidean Bianchi groups are conjugacy separable. As a consequence we deduce the result stated in the tittle.

scholarly journals Injectively immersed tori in branched covers over the figure eight knot

1993 ◽  
Vol 36 (2) ◽  
pp. 231-259
Author(s):  
Kerry N. Jones
Keyword(s):  
Fundamental Group ◽  
Decomposition Theorem ◽  
The Other ◽  
Group Level ◽  
Branched Covers ◽  
Graph Theoretic ◽  
Branched Cover ◽  
Figure Eight Knot

An algorithm is given for determining presence or absence of injectively (at the fundamental group level) immersed tori (and constructing them, if present) in a branched cover of S3, branched over the figure eight knot, with all branching indices greater than 2. Such tori are important for understanding the topology of 3-manifolds in light of (for example) the Jaco-Shalen–Johannson torus decomposition theorem and the fact that the figure eight knot is universal, i.e., that all 3-manifolds are representable as branched covers of S3, branched over the figure eight knot.The algorithm is principally geometric in its derivation and graph-theoretic in its operation. It is applied to two examples, one of which has an incompressible torus and the other of which is atoroidal.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

Grammatical Morpheme Intervention Issues for Students Who Use AAC

2008 ◽  
Vol 17 (2) ◽  
pp. 62-68 ◽  
Author(s):  
Cathy Binger
Keyword(s):  
Special Interest ◽  
Augmentative And Alternative Communication ◽  
Intervention Program ◽  
Annual Conference ◽  
Alternative Communication ◽  
Research Results ◽  
School Aged Children ◽  
Grammar Acquisition ◽  
Device Use

Abstract Many children who use AAC experience difficulties with acquiring grammar. At the 9th Annual Conference of ASHA's Special Interest Division 12, Augmentative and Alternative Communication, Binger presented recent research results from an intervention program designed to facilitate the bound morpheme acquisition of three school-aged children who used augmentative and alternative communication (AAC). Results indicated that the children quickly began to use the bound morphemes that were taught; however, the morphemes were not maintained until a contrastive approach to intervention was introduced. After the research results were presented, the conference participants discussed a wide variety of issues relating to grammar acquisition for children who use AAC. Some of the main topics of discussion included the following: provision of supports for grammar comprehension and expression, intervention techniques to support grammatical morpheme acquisition, and issues relating to AAC device use when teaching grammatical morpheme use.

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