Search

We study the relation between two uncountable groups with remarkable properties (cf. [15]): the topological free product of infinite cyclic groups G (the fundamental group of the Hawaiian Earring), and the inverse limit of finitely generated free groups F ˆ. The former has a canonical embedding as a proper subgroup of the latter and we examine when G , together with certain naturally defined normal subgroups of F ˆ generate the entire group F ˆ. We are interested in particular in normal subgroups Ker T (F ˆ) = ⋂ { Ker φ | φ ∈ hom ⁡ (F ˆ , T) } , where T is some finitely-presented n-slender group. Our main results state that if T is the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and Ker T (F ˆ) generate F ˆ. On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G ⋅ Ker T F ˆ is a proper subgroup of F ˆ. In the last section, we provide an interesting geometric interpretation of the above results in terms of path-connectedness of certain fibrations arising as inverse limits of covering spaces over the Hawaiian earring space. [ABSTRACT FROM AUTHOR]

Abstract The commutator subgroup of a free group is the subgroup consisting of elements contained in the kernel of every homomorphism from the free group to the integers. We give a similar characterization of the second derived subgroup of a free group. Specifically, we show that the second derived subgroup of a free group is equal to the intersection of the kernels of all homomorphisms into a solvable deficiency 1 group. [ABSTRACT FROM AUTHOR]

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R 3 . Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors. A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody. [ABSTRACT FROM AUTHOR]

Abstract: The mathematical model that Zermelo developed for ranking by paired comparisons and that was later popularized by Bradley and Terry has several attractive theoretical properties, but computation of the associated ratings may involve solution of a system of several high-degree polynomial equations in several variables. This paper describes how to define quantities analogous to electrical resistance and conductance for certain generalized tournaments in such a way that these quantities are well-behaved with respect to certain types of decomposition of tournaments and permit comparison of the ratings of pairs of nodes. Application of this theory is illustrated through consideration of specific examples. [Copyright &y& Elsevier]