Subgroups and Quotients of Fundamental Groups

Abstract

We explore the descriptive set theory of subgroups of fundamental groups, giving theorems regarding dichotomies on cardinality. In paticular, we show that the quotient of the fundamental group of a path connected, locally path connected Polish space by a normal subgroup which is sufficiently eay to describe topologically (of ``nice' pointclass having the property of Baire) is either countable or of cardinality continuum. In case the space is compact, countability of the quotient implies the quotient is finitely generated. We give upper bounds on the complexity of some subgroups, such as the shape kernel and the Spanier group. Applications to the normal generation of groups are given, as well as an application to covering space theory. We present an array of theorems regarding the first homology of Peano continua. We also demonstrate the existence of subgroups of all but finitely many additive and multiplicative Borel types. We prove that torsion-free word hyperbolic groups are n-slender, and the class of n-slender groups is closed under graph products.