Residual Finiteness Research Articles

Residual finiteness, QCERF and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.

Isometry groups of non-positively curved spaces: discrete subgroups

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

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.

Green index and finiteness conditions for semigroups

Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S ∖ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents.

“A Generalization of Residual Finiteness”

The concept of residual finiteness with respect to automorphic equivalence, a property generalizing residual finiteness and conjugacy separability is introduced. A sufficient condition for a group G to be residually finite with respect to automorphic equivalence is proven (Theorem). It is then used to give some examples of automorphic equivalent residually finite groups.

A generalization of residual finiteness

A generalization of residual finiteness

On the residual finiteness and other properties of (relative) one-relator groups

A relative one-relator presentation has the form P = (X, H ; R) where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form P (Theorem 6).

Reflections on the residual finiteness of one-relator groups

Let G = < a, b, … | r = 1 > be a one-relator group equipped with at least two generators. For all w which do not commute with r in the ambient free group on the generators a, b, …, the groups G(r,w) = < a,b, … | r r w = r 2 > are not residually finite and have the same finite images as G. The existence of this family of one-relator groups which are not residually finite reinforces what is becoming more obvious with time, that one-relator groups can be extremely complicated. This not only serves to underline the complexity of one-relator groups but provides us with the opportunity to raise a number of problems about these groups in the hope that they will stimulate further work on the conjugacy and isomorphism problems for one-relator groups as a whole.

Residual finiteness of outer automorphism groups of certain tree products

We prove that the outer automorphism groups of tree products of finitely many polycyclic-by-finite groups amalgamating central edge groups are residually finite. As a consequence, the outer automorphism groups of tree products of finitely many abelian groups are residually finite.

On superstable groups with residual properties

AbstractGiven a pseudovariety 𝒞, it is proved that a residually‐𝒞 superstable group G has a finite seriesG 0 ⊴ G 1 ⊴ · · · ⊴ G n = G such that G 0 is solvable and each factor G i +1/G i is in 𝒞 (0 ≤ i ≤ n – 1). In particular, a residually finite superstable group is solvable‐by‐finite, and if it is ω ‐stable, then it is nilpotent‐by‐finite. Given a finitely generated group G, we show that if G is ω ‐stable and satisfies some residual properties (residual solvability, residual finiteness, …), then G is finite. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x1,ny1] ⋯ [xm,nym])q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.

Residual finiteness of infinite amalgamated products of cyclic groups

We classify the residual finiteness of infinite amalgamated free products of infinite cyclic groups. As a side result we also classify residual finiteness and subgroup separability of the subgroups of ( Q , + ) .

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.

A strong form of residual finiteness for groups

A strong form of residual finiteness for groups

On finiteness conditions for Rees matrix semigroups

Let T=ℳ[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.

Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.

Ascending HNN extensions of polycyclic groups are residually finite

We prove that every ascending HNN extension of a polycyclic-by-finite group is residually finite. We also give a criterion for the residual finiteness of an ascending HNN extension of a residually nilpotent group, and apply this criterion to recover a result of Moldavanskiı̆ on the residual finiteness of certain ascending HNN extensions of free groups.

The residual finiteness of ascending HNN-extensions of certain soluble groups

If G is a group with an injective endomorphism φ, then the HNN-extension Gφ = 〈G, t : t−1gt = gφ for all g ∈ G〉 is called the ascending HNN-extension of G determined by φ. We prove that Gφ is residually finite when G is either finitely generated abelian-bypolycyclic-by-finite or reduced soluble-by-finite minimax. We also provide an example of a 3-generator residually finite soluble group G of derived length 3 with a non-residually-finite ascending HNN-extension.

Residual Finiteness of Quasi-Positive One-Relator Groups

Residual Finiteness of Quasi-Positive One-Relator Groups

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.