Some finitely presented nilpotent-by-abelian groups

We consider a subclass of the class of the nilpotent (of class 2)-by-abelian groups and classify the finitely presented groups in it.     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

Asymptotic dimension of finitely presented groups

We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.

     

JSJ-splittings for finitely presented groups over slender groups

We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually Z⊕Z groups. The approach, called “track zipping”, is relatively elementary, and differs from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called patterns) in a presentation 2-complex for the ambient group. Oblatum 18-IV-1997 & 30-I-1998 / Published online: 18 September 1998     

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

Subdirect products of finitely presented metabelian groups

There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability.We show that, in a sense that can be made precise, ‘most’ subdirect products of a finite set of finitely presented metabelian groups are again finitely presented. To be a little more precise, we assign to each subdirect product a point of an algebraic variety and show that, in most cases, those points which correspond to non-finitely presented subdirect products form a subvariety of smaller dimension.     

Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.     

Preservation and distortion of area in finitely presented groups

IfK=G where is a tame automorphism of the 1-relator groupG, then the combinatorial area of loops in a Cayley graph ofG is undistorted in a Cayley graph ofK. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental group of a compact 3-manifold does not distort area.Partially supported by NSF grant DMS-9200433.     

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X   such that _b2(X)?1$b_2(X)?1$ we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G  , with prescribed values on certain loops. If _b2(X)=1$b_2(X)=1$ we assume G=SO(3)$G=\mathrm{SO}(3)$ and that the cup product on H¹(X;Q)$H^1(X;\mathbb{Q})$ is non-zero.