1. CMB 2017 (vol 60 pp. 830)
||Generalized Torsion Elements and Bi-orderability of 3-manifold Groups|
It is known that a bi-orderable group has no generalized torsion
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
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)
||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
Keywords:Burnside groups, free products of groups, group rings, zero-divisors
Categories:20C07, 20E06, 20F05, , 20F50
3. CMB 2011 (vol 56 pp. 395)
||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)
||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)
6. CMB 2003 (vol 46 pp. 299)
7. CMB 1998 (vol 41 pp. 231)
||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