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Search: MSC category 20F05 ( Generators, relations, and presentations )

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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 2012 (vol 57 pp. 326)

Ivanov, S. V.; Mikhailov, Roman
On Zero-divisors in Group Rings of Groups with Torsion
Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

Keywords:Burnside groups, free products of groups, group rings, zero-divisors
Categories:20C07, 20E06, 20F05, , 20F50

3. CMB 2011 (vol 56 pp. 395)

Oancea, D.
Coessential Abelianization Morphisms in the Category of Groups
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi$ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews--Curtis conjecture.

Keywords:coessential epimorphism, Nielsen transformations, Andrew-Curtis transformations
Categories:20F05, 20F99, 20J15

4. CMB 2006 (vol 49 pp. 347)

Ecker, Jürgen
Affine Completeness of Generalised Dihedral Groups
In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every $k \in~\N$ the $k$-affine complete generalised dihedral groups. We find that the direct product of a $1$-affine complete group with itself need not be $1$-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for $2$-affine completeness.

Categories:08A40, 16Y30, 20F05

5. 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

6. CMB 2003 (vol 46 pp. 299)

Tomaszewski, Witold
A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group
We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

Categories:20E05, 20F12, 20F05

7. CMB 1998 (vol 41 pp. 231)

Worthington, R. L.
The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators
The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators are explicitly calculated. The assertions of J.~Cannon and Ph.~Wagreich for the 4-generated groups, that the poles of the growth series lie on the unit circle, with the exception of a single real reciprocal pair of poles, are verified. We also verify that for the 5-generated groups, this phenomenon fails.

Categories:20F05, 20F55

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