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.

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.

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.

We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

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.