The Combinatorial Structure of the Hawaiian earring group. (with Jim Cannon)
Abstract: We show how to represent the Hawaiian earring group as a group of transfinite words on a countably infinite alphabet exactly analogous to the representation of a finite rank free group as finite words on a finite alphabet. We define big free groups similarly as the groups of transfinite words on sets of arbitrary cardinality, and study their group theoretic properties. To appear in Topology and its Applications.
On the Fundamental Groups of One-Dimensional Spaces. (with Jim Cannon)
Abstract: We show various properties of the fundamental groups of generic topological spaces and in particular one-dimensional spaces. The main result is that the fundamental group of a second countable, connected, locally path connected one-dimensional metric space is free if and only if it is countable, if and only if the space has a universal cover.
The Big Fundamental Group, Big Hawaiian Earrings and the Big Free Groups. (with Jim Cannon)
Abstract: We describe the notion of big fundamental group. Informally, this is the group one gets by replacing the unit interval in the notions of arc and homotopy by arbitrary compact, connected ordered topological spaces (which are called big arcs) . A big Hawaiian Earring is the one-point compactification of a disjoint union of open intervals. We show that the big fundamental group of a big Hawaiian earring is a big free group. To appear in Topology and its Applications.
On the fundamental groups of planar sets. (with Jack Lamoreaux)
Abstract: In this paper we prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalencies to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article which states that a connected, locally path connected subset of the Euclidean plane admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.
Discreteness properties of translation numbers in solvable groups.
Abstract: We define a group to be translation proper if it carries a left-invariant metric in which the translation numbers of the non-torsion elements are nonzero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension, m, is a finite extension of the free abelian group of rank m.
Translation Numbers of Groups acting on Quasiconvex Spaces.
Abstract: We define a group to be translation discrete if it carries a metric in which the translation numbers of the non-torsion elements are bounded away from zero. We define the notion of quasiconvex space which generalizes the notion of both CAT(0) and Gromov--hyperbolic spaces. We show that a cocompact group of isometries acting properly discontinuously cocompactly on a proper quasiconvex metric space is translation discrete if and only if it does not contain an essential Baumslag-Solitar quotient. It follows that if such a group is either biautomatic or residually finite then it is translation discrete.