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Search: MSC category 57M05 ( Fundamental group, presentations, free differential calculus )

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1. CMB Online first

Motegi, Kimihiko; Teragaito, Masakazu
Generalized torsion elements and bi-orderability of 3-manifold groups
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of $3$-manifolds, and verify the conjecture for non-hyperbolic, geometric $3$-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic $3$-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2, m)$ ($m \gt 2$) is a generalized torsion element.

Keywords:generalized torsion element, bi-ordering, 3-manifold group
Categories:57M25, 57M05, 06F15, 20F05

2. CMB 2010 (vol 53 pp. 706)

Roberts, R.; Shareshian, J.
Non-Right-Orderable 3-Manifold Groups
We exhibit infinitely many hyperbolic $3$-manifold groups that are not right-orderable.

Categories:20F60, 57M05, 57M50

3. CMB 2003 (vol 46 pp. 310)

Wang, Xiaofeng
Second Order Dehn Functions of Asynchronously Automatic Groups
Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Keywords:second order Dehn function, combing, asynchronously automatic group
Categories:20E06, 20F05, 57M05

4. CMB 2002 (vol 45 pp. 231)

Hironaka, Eriko
Erratum:~~The Lehmer Polynomial and Pretzel Links
Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451.

Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
Categories:57M05, 57M25, 11R04, 11R27

5. CMB 2002 (vol 45 pp. 131)

Przytycki, Józef H.; Rosicki, Witold
The Topological Interpretation of the Core Group of a Surface in $S^4$
We give a topological interpretation of the core group invariant of a surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of $S^4$ with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

Categories:57Q45, 57M12, 57M05

6. CMB 2001 (vol 44 pp. 440)

Hironaka, Eriko
The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
Categories:57M05, 57M25, 11R04, 11R27

7. CMB 1999 (vol 42 pp. 257)

Austin, David; Rolfsen, Dale
Homotopy of Knots and the Alexander Polynomial
Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial.

Categories:57N10, 57M05, 57M25, 57N65

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