Ralph Strebel

Almost finitely presented soluble groups

1.1 Finitely presented soluble groups have been investigated by several authors. Roughly speaking, their results deal with two aspects: with the subgroup structure of finitely presented soluble groups [2], [4], [7], [18], and with soluble varieties whose non-cyclic relatively free group are infinitely related [16], [3]. Here we attack the problem of recognizing which finitely generated soluble groups are finitely related in a somewhat more systematic way: We show that all finitely presented soluble groups have a certain structural property which is inherited by homomorphic images (whether this holds for the property of being finitely presented itself is an old problem of P. Halls and is still open). 1.2 The methods of [2] and [4] made it clear that even in the soluble case the HNN-construction is an important tool for obtaining finitely presented groups. Recall that every group B containing a pair of isomorphic subgroups 0 : S---~ T is embedded in the HNN-group

A geometric invariant for nilpotent-by-abelian-by-finite groups

We attach to every finitely generated nilpotent-by-Abelian-by-finite group G a closed subset σ(G) of a sphere and demonstrate that this geometric invariant measures certain finiteness properties of G, such as being finitely presented, constructible, coherent, or polycyclic-by-finite.

On Groups of PL-homeomorphisms of the Real Line

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompsons group F thus: G consists of all orientation preserving PL-homeomorphisms of the real line with supports in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive P submodule A of R. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Main aims of the monograph are the determination of isomorphisms among the groups G and the study of their automorphism groups. Complete answers are obtained if the group P is not cyclic or if the interval I is the full line.

Some finitely generated, infinitely related metabelian groups with trivial multiplicator

Abstract In this article we study the multiplicator and the number of relators of some metabelian groups that arise from specially simple one-relator groups by the process of metabelianization. This leads, in particular, to the seemingly simplest example of a finitely generated, but infinitely related group with trivial multiplicator.

Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

On homological duality

Abstract Duality relations given in terms of natural isomorphisms Tor R n−j (D, -) → ∼ Ext j R (C, -) are shown to be related with a straightforward generalization of the operation of taking the dual HomR(P, R) of a finitely generated projective module P. The obtained results allow to characterize duality groups (in the sense of Bieri-Eckmann) and inverse duality groups in a perspicuous way, and to generalize two main result of Matlis [13].

Explicit resolutions for the binary polyhedral groups and for other central extensions of the triangle groups

This paper will exhibit two classes of finitely presented groups for which explicit free resolutions can be obtained by direct algebraic calculations. The resolutions will be either periodic of period 4, or of length 3. 1. The groups of the first class are central extensions of the triangle groups, admitt ing a 2-genera tor 2-re la tor presentation. Specifically, let l, m, n be integers with min (l/I, [ml, [n[)>=2 and define

On one-relator soluble groups

and ~ a soluble variety. Following G. Baumslag we call G~(G), where ~(G) denotes the ~-verbal subgroup of G, a one-relator ~-group. G. Baumslag initiated the search for conditions on m, w and ~ that force G/~(G) to be infinitely related. He showed this to be the case if ~ = 9X 2 (i.e. the variety of all metabelian groups) and m ->_ 3, w arbitrary, or m = 2, w a proper power ([2], p. 67, Theorem F). These results were extended in [4] (p. 259, Theorem B): If m >_- 3 and ~3 is a soluble variety containing a subvariety of the form 91p �9 9~, p a prime, then G/~(G) is infinitely related. This result is actually best possible; for a soluble variety

Fitting quotients of finitely presented abelian-by-nilpotent groups

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.