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Craig R. Guilbault

Publications and source records attributed to Craig R. Guilbault.

17 recordsLinked to original sources
arXiv2023

A generalized theory of expansions and collapses with applications to Z-compactification

We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of compactification theorems. We are particularly interested in Z-set compactifications, which play an important role in geometric group theory and in algebraic and geometric topology.

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arXiv2022

Group boundaries for semidirect products with Z

Bestvina's notion of a Z-structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an EZ-structure has proven useful in attacks on the Novikov Conjecture and related problems. Characterizations of groups admitting a Z- or EZ-structure are longstanding open problems. In this paper, we examine semidirect products of a group G with the integers. For example, we show that, if G is torsion-free and admits a Z-structure, then so does every semidirect product of this type. We prove a similar theorem for EZ-structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit Z-structures, as do all strongly polycyclic groups, and all groups of polynomial growth. In those latter two cases our Z-boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an EZ-structure and discuss new applications to the Novikov Conjecture.

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arXiv2021

Ends, shapes, and boundaries in manifold topology and geometric group theory

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard tools of algebraic topology. One approach uses end invariants, such as the number of ends or the fundamental group at infinity; another approach seeks nice compactifications, then analyzes the boundaries. A thread connecting the two is shape theory. In these notes we provide a careful development of several topics: homotopy and homology properties and invariants for ends of spaces, proper maps and homotopy equivalences, tameness conditions, shapes of ends, and various types of Z-compactifications and Z-boundaries. Classical and current research from both manifold topology and geometric group theory provide the context. Along the way, many open problems are encountered. Our primary goal is casual but coherent introduction that is accessible to graduate students and also of interest to active mathematicians whose research might benefit from knowledge of these topics.

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arXiv2021

Coarse $\mathcal{Z}$-Boundaries for Groups

We generalize Bestvina's notion of a $\mathcal{Z}$-boundary for a group to that of a "coarse $\mathcal{Z}$-boundary." We show that established theorems about $\mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $\mathcal{Z}$-boundaries become theorems when applied to coarse $\mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $\mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a "model $\mathcal{Z}$-geometry." In accordance with the existing theory, we also develop an equivariant version of the above -- that of a "coarse $E\mathcal{Z}$-boundary."

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arXiv2020

Extreme Nonuniqueness of End-Sum

We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding the nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of the summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

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arXiv2018

Compactifications of manifolds with boundary

This paper is concerned with "nice" compactifications of manifolds. Siebenmann's iconic dissertation characterized open manifolds M^m (m>5) compactifiable by addition of a manifold boundary. His theorem extends easily to cases where M^m is noncompact with compact boundary; however, when Bd(M^m) is noncompact, the situation is more complicated. The goal becomes a "completion" of M^m, ie, a compact manifold C^m and a compact subset A such that C^m\A = M^m. Siebenmann did some initial work on this topic, and O'Brien extended that work to an important special case. But, until now, a complete characterization had yet to emerge. We provide such a characterization. Our second main theorem involves Z-compactifications. An open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann guarantee Z-compactifiability for a manifold M^m. We cannot answer that question, but we do show that those conditions are satisfied if and only if M x [0,1] is Z-compactifiable. A key ingredient is the above Manifold Completion Theorem---an application that partly explains our current interest in that topic, and also illustrates the utility of the conditions found in that theorem.

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arXiv2018

Boundaries of Baumslag-Solitar Groups

A $\mathcal{Z}$-structure on a group $G$ was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $\mathcal{EZ}$-structure. The general questions of which groups admit $\mathcal{Z}$- or $\mathcal{EZ}$-structures remain open. In this paper we add to the current knowledge by showing that all Baumslag-Solitar groups admit $\mathcal{EZ}$-structures and all generalized Baumslag-Solitar groups admit $\mathcal{Z}$-structures.

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arXiv2018

Proper homotopy types and Z-boundaries of spaces admitting geometric group actions

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups with torsion. The main theorems are new in that they generalize results found in the literature, but a significant aim is expository. Toward that end, brief but reasonably comprehensive introductions to the theories of ANRs (absolute neighborhood retracts) and Z-sets are included, as well as a much shorter short introduction to shape theory. Here is a sampling of the theorems proved here. THEOREM. If quasi-isometric groups G and H act geometrically on proper metric ARs X and Y , resp., then X is proper homotopy equivalent to Y. THEOREM. If quasi-isometric groups G and H act geometrically on proper metric ARs X and Y , resp., and Y can be compactified to a Z-structure for H, then the same boundary can be added to X to obtain a Z-structure for G. THEOREM. If quasi-isometric groups G and H admit Z-structures (X^, Z_1) and (Y^, Z_2) resp., then Z_1 and Z_2 are shape equivalent.

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arXiv2015

Noncompact manifolds that are inward tame

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. We show that these manifolds have 'almost perfectly semistable' fundamental group at each end. That observation leads to further analysis of group theoretic conditions at infinity, and to the notion of a 'near pseudo-collar' structure. We obtain a complete characterization of n-manifolds (n>5) admitting such a structure, thereby generalizing earlier work. We also construct examples illustrating the necessity and usefulness of new conditions introduced here. Variations on the notion of a perfect group, with corresponding versions of the Quillen Plus Construction, form an underlying theme.

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arXiv2015

A Comparison of Large Scale Dimension of a Metric Space to the Dimension of its Boundary

Buyalo and Lebedeva have shown that the asymptotic dimension of a hyperbolic group is equal to the dimension of the group boundary plus one. Among the work presented here is a partial extension of that result to all groups admitting $\mathcal{Z}$-structures; in particular, we show that $\hbox{asdim}G\geq \hbox{dim}Z+1$ where $Z$ is the $\mathcal{Z}$-boundary.

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arXiv2013

Weak Z-structures for some classes of groups

Motivated by the usefulness of boundaries in the study of hyperbolic and CAT(0) groups, Bestvina introduced a general approach to group boundaries via the notion of a Z-structure on a group G. Several variations on Z-structures have been studied and existence results have been obtained for some very specific classes of groups. However, little is known about the general question of which groups admit any of the various Z-structures, aside from the (easy) fact that any such G must have type F, i.e., G must admit a finite K(G,1). In fact, Bestvina has asked whether every type F group admits a Z-structure or at least a "weak" Z-structure. In this paper we prove some rather general existence theorems for weak Z-structures. Among our results are the following: Theorem A. If G is an extension of a nontrivial type F group by a nontrivial type F group, then G admits a Z-structure. Theorem B. If G admits a finite K(G,1) complex K such that the corresponding G-action on the universal cover contains a non-identity element properly homotopic to the identity, then G admits a weak Z-structure. Theorem C. If G has type F and is simply connected at infinity, then G admits a weak Z-structure.

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arXiv2013

On the dimension of Z-sets

We offer a short and elementary proof that, for a Z-set A in a finite-dimensional ANR Y, dimA<dimY. This result is relevant to the study of group boundaries. The original proof by Bestvina and Mess relied on cohomological dimension theory.

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arXiv2012

Spherical alterations of handles: embedding the manifold plus construction

A key tool in our earlier work on ends of manifolds high-dimensional manifolds was an ability to embed cobordisms provided by the Quillen Plus Construction into those ends. Here we develop a `spherical modification' trick which provides a constructive approach to obtaining such embeddings. More importantly, this approach allows for more general embedding results. In this paper we develop generalizations of the plus construction and show how the corresponding cobordisms can be embedded in manifolds satisfying appropriate fundamental group properties. Results obtained here play an important role in our ongoing study of noncompact manifolds.

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arXiv2011

Topological properties of spaces admitting free group actions

In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse sequence of finitely generated free groups. We improve upon this latter result, by showing that X must have a stable finitely generated free fundamental group at infinity. Simple examples show that a free group of any finite rank is possible. We also prove that if X (as above), admits a non-cocompact action of Z+Z by covering transformations, then X is simply connected at infinity. Corollary: Every finitely presented one-ended group G which contains an element of infinite order satisfies exactly one of the following: 1) G is simply connected at infinity; 2) G is virtually a surface group; 3) The fundamental group at infinity of G is not pro-monomorphic. Our methods also provide a quick new proof of Wright's open manifold theorem.

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arXiv2006

Products of open manifolds with R

In this note we present a characterization of those open n-manifolds (n>4), whose products with the real line are homeomorphic to interiors of compact (n+1)-manifolds with boundary.

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arXiv2005

A solution to de Groot's absolute cone conjecture

A compactum X is an `absolute cone' if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n<4 and false for n>4. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincare Conjecture is true.

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