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An identity theorem for multi-relator groups, Math. Proc. Camb. Phil. Soc. 109 (1991) 313-321. MR92c: 20056
The Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included is a family of two-relator groups that was introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology calculations and a classification of finite subgroups.
Aspherical relative presentations, Proc. Edinburgh Math. Soc. 35 (1992) 1-39. MR93d: 57019
A geometric hypothesis is presented under which the cohomology of a group G given by generators and defining relators can be computed in terms of a group H definied by a subpresentation. In the presence of this hypothesis, which is framed in terms of spherical pictures, one has that H is naturally embedded in G, and that the finite subgroups of G are determined by those of H. Practical criteria for the hypothesis to hold are given. The theory is applied to give simple proofs of results of Collins-Perraud and of Kanevskii. In addition, we consider in detail the situation where G is obtained from H by adjoining a single new generator x and a single defining relator of the form axbxcx^n where |n| = 1.
Mayer-Vietoris sequences homotopy of 2-complexes and in homology of groups, J. Pure Appl. Algebra 77 (1992) 39-65. MR93e: 20069
This paper is concerned with the second homotopy groups (both absolute and relative) of unions of CW complexes. Sufficient homological conditions are given under which the homotopy modules of a union admit algebraic decompositions in terms of the homotopy properties of the summands. For two-complexes, these sufficient conditions are also necessary in the relative case Applications are given to the homology of groups. An eight-term exact sequence is constructed that relates the integral homology of a group F/RS to the integral homology of F/R and F/S. Computable representations of the thrid integral homology are introduced using equators of spherical maps. Many examples and computations are given.
J. H. C. Whitehead's asphericity question, in: Two-Dimensional Homotopy and Combinatorial Group Theory, C. Hog-Angeloni, W. Metzler, and A. J. Sieradski, editors, London Math. Soc. Lecture Note Series 197 (Cambridge University Press, 1993) 309-334. MR95g: 57006
This article is concerned with a question posed by Whitehead in1941: ``Is any subcomplex of an aspherical, 2-dimensional complex itself aspherical?'' This question remains unanswered despite considerable expense of effort; a wide variety of results is scattered throughout the literature. The present intent is to survey these efforts and to present both a summary of the published results and an overview of the methods that have been used in the study of the problem.
Calculating generators of pi2,
in: Two-Dimensional Homotopy and Combinatorial Group Theory, C. Hog-Angeloni, W. Metzler, and A. J. Sieradski editors, London Math. Soc. Lecture Note Series 197 (Cambridge University Press, 1993) 157-188 (with S. J. Pride). MR95g: 57006
This article discusses combinatorial geometric techniques that determine explicit generators for the second homotopy module of a two-complex in terms of its cell structure. Applications of the techniques are also presented. The discussion focuses on the theory of pictures. The first section provides an overview of the theory of pictures from a homotopy-theoretic perspective. The second section deals with generalities related to the generation of second homotopy modules. The third section is devoted to a summary description without proofs of various calculations and applications that have been obtained in the study of second homotopy modules.
Unions of Cockcroft two-complexes, Proc. Edinburgh Math. Soc. 37 (1994) 317-324. MR95e: 57005
A combinatorial group-theoretic hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups given by generators and relators.
A group-theoretic reduction of J. H.
C. Whitehead's asphericity question, in: Groups-Korea '94, A. C. Kim and D. L. Johhnson, editors, (de Gruyter, 1995) 15-24. MR98f:57005
J. H. C. Whitehead asked in 1941 whether subcomplexes of aspherical two-complexes are aspherical. The question remains unanswered as of this writing. In this note we use a theorem of J. Howie to show that Whitehead's question can be reduced to two problems in combinatorial group theory. Some partial results are surveyed.
On the asphericity of length four relative group presentations, Internat. J. Algebra Comput. 7 (1997) 277--312
Excluding five unresolved cases, asphericity is classified in relative group presentations of the form (H,x: xaxbxcxd).
Second order Dehn functions of groups, Quart. J. Math. Oxford,
Ser. (2) 49 (1998) 1-30.
In this paper we compute the second order Dehn function for various classes of groups. The second order Dehn function provides an isoperimetric measure of the algebraic, combinatorial, and geometric complexity of the second homotopy module associated to a group presentation (or two-complex). We display an infinite variety of subquadratic growth types for these functions. Word hyperbolic groups are shown to have linear second order Dehn function. Particular attention is paid to the behavior of these functions with respect to subgroups and also direct products of groups.
Splittings of homotopy modules and normalizers in groups, preprint. (25 pages)
This paper relates second homotopy modules to normalizers in
groups. The results can be phrased in terms of pairs of group presentations or
in terms of pairs of connected two-complexes. There has been a great deal of recent work on the general problem of determining generators for second homotopy modules. We use some of this work as motivation and as material for examples. In addition, our work begins an investigation into the internal structure of these modules.
Cockcroft properties
of Thompson's group, submitted for publication.
In a study of the word problem for groups, R. J.
Thompson considered a certain group $F$ of self-homeomorphisms of the
Cantor set and showed, among other things, that $F$ is finitely
presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer
showed that $F$ is the fundamental group of a finite two-complex
$Z^{2}$ having Euler characteristic one and which is Cockcroft,
in the sense that each map of the two-sphere into $Z^{2}$ is
homologically trivial. We show that no proper covering complex of
$Z^{2}$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
two-complex with fundamental group $F$ is Cockcroft.
Weighted combinatorial group
theory and wild metric complexes, submitted for publication.
In this paper, we develop the low dimensional homotopy theory required for
weighted combinatorial group theory. In Omega-groups, an
earlier paper by Sieradski, the usual concepts of generators and relators of
group presentations are extended to weighted generators and weighted relators
for weighted group presentations. This extension parallels the passage from
finite sets to order types, i.e. closed nowhere dense sets in the closed unit
interval. In the weighted environment, products of all order-type are
permitted, provided that the entries of the product have weights that limit at
zero as their depth of occurrence in the order-type increases without bound.
Here, we develop weighted analogs of the usual correspondence via fundamental
groups between free groups and 1-dimensional CW cell complexes and between
group presentations and 2-dimensional CW cell complexes. The results are a
correspondence between free omega-groups and wild metric
1-complexes in which the 1-cells can limit on 0-cells and a
correspondence between weighted group presentations and wild
metric 2-complexes in which the 1-cells and 2-cells can limit on the
0-cells.
Universal path spaces, submitted for publication.
This paper examines a theory of universal path spaces that properly
includes the covering space theory of connected, path connected, semi-locally
simply connected spaces. The latter hypothesis that each point in the base
space has a relatively simply connected open neighborhood---necessary and
sufficient for the existence of a simply connected covering space---is
abandoned, thus admitting as base even those spaces that contain arbitrarily
small essential loops at wild points. When the base space is a wild metric
2-complex, the universal path space is simply connected if and only if
the fundamental group is an omega-group--a group whose elements
acquire non-negative real weights and form countable products of all
order-type whenever their weights vanish as their appearance in the order-type
deepens. Then the endpoint projection enjoys all of the standard lifting
properties of covering projections; in particular, the fundamental group of
the base space is isomorphic to the group of equivariant self-homeomorphisms
of the universal path space. But the properly discontinuous action of the
fundamental group on the discrete fibers over the tame
points of the base space converge to continuous actions on Cantor
fibers (i.e., totally disconnected and perfect ones) over the wild
points of the base space. The standard features of covering space theory
are thus engulfed by the richer features of universal path space theory for a
variety of base spaces whose wild local topology prevents the application of
traditional covering space theory.