In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, P-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numb...

A continuous function p : S̃ → S between two surfaces S̃ and S is called a branched covering if there exists a finite set B in S such that the restriction of p to S̃ − p−1(B), p | S̃−p−1(B) : S̃− p −1(B)→ S− B, is a covering projection in the usual sense. The smallest set B of S which has this property is called the branch set. A branched covering p : S̃ → S is regular if there exists a (finite) grou...

Subgroup growth is an important new area of group theory. It attempts to quantify the number of finite index subgroups of a group, as a function of their index. In this paper, we will provide new, strong lower bounds on the subgroup growth of a variety of different groups. This will include the fundamental groups of all finite-volume hyperbolic 3-manifolds. By using the correspondence between s...

An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeomorphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine transfo...

There is much interest in finding short presentations for the finite simple groups. In a previous paper we produced nice efficient presentations for all small simple groups and for their covering groups. Here we extend those results from simple group order less than 100,000 up to order one million, but we leave one simple group and one covering group for which the efficiency question remains un...