The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

Full residual finiteness growths of nilpotent groups

Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely n^b, where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form n^b where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

     

Groups with finiteness conditions on the lower central series of non-normal subgroups

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group G the subgroup \(\gamma _{k}(G)\) is finite if the set \(\{\gamma _{k}(H)\;|\;H\le G,\,H\ntriangleleft G\}\) is finite. Moreover, locally graded groups with finitely many kth terms of lower central series of infinite non-normal subgroups are also completely described.     

On residual finiteness of direct products of algebraic systems

It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.      

On the residual finiteness of descending HNN-extensions of groups

Let G be a group of finite generic rank, φ an injective endomorphism of the group G, and G(φ) the descending HNN-extension of G corresponding to the endomorphism φ. Let the index of the subgroup Gφ in G be finite and equal to n. It is proved that, if the group G is almost residually π-finite for some set π of primes coprime to n, then the group G(φ) is residually finite. This generalizes a series of known results, including the Wise-Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

Faithful linear representations and residual finiteness of some products of cycles with one relation

Dortmund, Germany. Novosibirsk, USSR. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 204–206, January–February, 1991.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.