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.

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

Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated.

We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups.

Using the theory of $p$-adic Lie groups we give conditions for a finitely generated group to admit a splitting as a non-trivial free product with amalgamation. This can be viewed as an extension of a theorem of Bass.

We show that if $A$ is a torsion-free word hyperbolic group which belongs to class $(Q)$, that is all finitely generated subgroups of $A$ are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns subgroup of $A$. This, in particular, implies that if $B$ is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting $3$-manifold groups are known to belong to class $(Q)$ and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.