Subgroup properties of fully residually free groups

We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.

     

Subgroup separability of certain generalized free products of nilpotent-by-finite groups

In this paper, we show that certain generalized free products of nilpotent-by-finite groups are subgroup separable when the amalgamated subgroup is h× D where D is in the center of both factors.     

Subgroup separability and virtual retractions of groups

We discuss separability properties of discrete groups, and introduce a new property of groups that is motivated by a geometric proof of separability of geometrically finite subgroups of Kleinian groups. This property appears natural in that it provides a general framework for old questions in the geometry and topology of hyperbolic manifolds and discrete groups.     

Subgroup separability of graphs of abelian groups

In the present paper we give necessary and sufficient conditions for the subgroup separability of the fundamental group of a finite graph of groups with finitely generated abelian vertex groups.

     

Economical separability in free groups

Consider the rank n free group F _n with basis X. Bogopol’ski? conjectured in [1, Problem 15.35] that each element w ∈ F _n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F _n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F _n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.     

Subgroup separability, knot groups and graph manifolds

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let denote the fundamental group of the link consisting of a chain of circles. It is shown that is not subgroup separable. Furthermore, it is shown that is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain .

     

The isomorphism problem for finitely generated fully residually free groups

We prove that the isomorphism problem for finitely generated fully residually free groups (or F-groups for short) is decidable. We also show that each F-group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms .     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

Commutators in residually finite groups

The following result is proved. Let n be a positive integer and G a residually finite group in which every product of at most 68 commutators has order dividing n. Then G′ is locally finite.     

Centralizers in residually finite torsion groups

Let be a residually finite torsion group. We show that, if has a finite 2-subgroup whose centralizer is finite, then is locally finite. We also show that, if has no -torsion, and is a finite 2-group acting on in such a way that the centralizer is soluble, or of finite exponent, then is locally finite.

     

Magnus pairs in,and free conjugacy separability of,limit groups

We study the \(\mathbb {Z}\)-module of first order local Vassiliev type invariants of stable maps of oriented 3-manifolds into \(\mathbb {R}^4\). As a previous step, we determine a complete classification of the codimension two germs and multigerms as well as their corresponding bifurcation diagrams. This allows us to show the existence of 4 generators for this module. In particular, we see that the number of pairs of quadruple point, the number of transversal intersections of the crosscap suspension with immersive branches and the Euler number of the image are first order local Vassiliev type invariants for these maps. We also prove that the total number of connected components of the triple point curve is a non local Vassiliev type invariant.     

Multilinear commutators in residually finite groups

The following result is proved. Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.     

On certain residually finite groups

In this paper we will give necessary and sufficient conditions under which A ⊕ B = A ⊕ C implies B and C are comparable relative to ≤ for all finitely generated projective modules A, B and C over a regular ring     

Subgroup separability of the figure 8 knot group

It is shown that the fundamental groups of certain non-positively curved 2-complexes have the property that their quasiconvex subgroups are the intersections of finite index subgroups.As a consequence, every geometrically finite subgroup of the figure 8 knot group is the intersection of finite index subgroups. The same result holds for many other prime alternating link groups.     

A footnote on residually finite groups

We give an easy proof that a finitely generated group which is residually (finite and soluble of bounded rank) is nilpotent by quasi-linear. This can be used to shorten the proofs of some recent theorems about residually finite groups.