Gregory R. Conner

The combinatorial structure of the Hawaiian earring group

Abstract In this paper we study the combinatorial structure of the Hawaiian earring group, by showing that it can be represented as a group of transfinite words on a countably infinite alphabet exactly analogously to the representation of a finite rank free group as finite words on a finite alphabet. We define a big free group similarly as the group of transfinite words on given set, and study their group theoretic structure.

One-dimensional sets and planar sets are aspherical

Abstract We give a relatively short proof of the theorem that planar sets are aspherical. The first proof of this theorem, by third author Andreas Zastrow, was considerably longer.

The big fundamental group, big Hawaiian earrings, and the big free groups ✩

Abstract In this second paper in a series of three we generalize the notions of fundamental group and Hawaiian earring. In the first paper we generalized the notion of free group to that of a big free group . In the current article we generalize the notion of fundamental group by defining the big fundamental group of a topological space. We also describe big Hawaiian earrings , which are generalizations of the classical Hawaiian earring. We then prove that the big fundamental group of a big Hawaiian earring is a big free group.

An extension of Zermelo's model for ranking by paired comparisons

In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo’s model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use. Suppose that n objects are compared a pair at a time, and that for each comparison one of the two objects in the pair is judged superior to the other. (A common example would be athletic teams engaged in pairwise competitions.) The results of the comparisons can be summarized in the outcome matrix A =( aij), where aij is the number of comparisons in which object i is judged to be superior to object j. If all of the o-diagonal elements in A +A T are the same (i.e. there has been roundrobin competition) then the natural way to rank objects would be according to their scores si = P n=1aij. If, on the other hand, the outcome matrix lacks this symmetry, it is reasonable to suspect that ranking by score is not necessarily the best possible choice. (Contrary to common mathematical usage, we will often use the word tournament when referring to this asymmetric case; when we wish to emphasize the possible lack of symmetry, we may use the phrase generalized tournament.) A wide variety of methods have been proposed for ranking generalized tournaments. (See, for example, [9, 10].) In 1929, Zermelo [33] derived the functional

The fundamental group of a visual boundary versus the fundamental group at infinity

Abstract There is a natural homomorphism from the fundamental group of the boundary of any non-positively curved geodesic space to its fundamental group at infinity. We will show that this homomorphism is an isomorphism in case the boundary admits a universal covering space, and that it is injective in case the boundary is one-dimensional.

Commensurated subgroups, semistability and simple connectivity at infinity

A subgroup Q of a group G is commensurated if the commensurator of Q in G is the entire group G . Our main result is that a finitely generated group G containing an infinite, finitely generated, commensurated subgroup H of infinite index in G is one-ended and semistable at1. Furthermore, if Q and G are finitely presented and either Q is one-ended or the pair .G;Q/ has one filtered end, then G is simply connected at1. A normal subgroup of a group is commensurated, so this result is a generalization of M Mihalik’s result [17] and of B Jackson’s result [11]. As a corollary, we give an alternate proof of V M Lew’s theorem that a finitely generated group G containing an infinite, finitely generated, subnormal subgroup of infinite index is semistable at1. So several previously known semistability and simple connectivity at1 results for group extensions follow from the results in this paper. If W H ! H is a monomorphism of a finitely generated group and . H/ has finite index in H , then H is commensurated in the corresponding ascending HNN extension, which in turn is semistable at1. 20F69; 20F65

Local topological properties of asymptotic cones of groups

We define a local analogue to Gromov’s loop division property which we use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. When considering groups our condition allows us to relate the local connectedness properties of the asymptotic cone with combinatorial properties of the group. This is used to understand the asymptotic cones of many groups actively being studied in the literature. 20F65; 20F69

The geometry and fundamental groups of solenoid complements

A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with nonhomeomorphic complements.

Neighborhood monotonicity, the extended Zermelo model, and symmetric knockout tournaments

In this paper, neighborhood monotonicity is presented as a natural property for methods of ranking generalized tournaments (directed graphs with weighted edges). An extension of Zermelos classical method of ranking tournaments is shown to have this property. An estimate is made of the proportion of ordered pairs that all neighborhood-monotonic rankings of symmetric knockout tournaments have in common. Finally, numerical evidence for the asymptotic behavior of the extended Zermelo ranking of symmetric knockout tournaments is presented.

Inverse limits of finite rank free groups

Abstract. We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.