There is a well-known correspondence between infinite trees and ultrametric spaces that comes from considering the end space of the tree. The correspondence is interpreted here as an equivalence between two categories, one of which encodes the geometry of trees at infinity and the other encodes the micro-geometry of complete ultrametric spaces.

The ends of a topological space are the directions in which it becomes non-compact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behaviour at infinity of a non-compact space. The second part studies tame ends in topology. Tame ends are shown to have a uniform structure, with a periodic shift map. Approximate fibrations are used to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory.

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h-cobordism extension theorem.

Abstract. The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths which lie in a single stratum except possibly at their initial points form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Abstract. Hughes has defined a class of groups, which we call FSS (finite similarity structure) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson's group V .Guided by previous work on Thompson's group V , we establish a number of new results about FSS groups. Our main result is that a class of FSS groups are of type F∞. This generalizes work of Ken Brown from the 1980s. Next, we develop methods for distinguishing between isomorphism types of some of the Nekrashevych-Röver groups V d (H), where H is a finite group, and show that all such groups V d (H) have simple subgroups of finite index. Lastly, we show that FSS groups defined by small Sim-structures are braided diagram groups over tree-like semigroup presentations. This generalizes a result of Guba and Sapir, who first showed that Thompson's group V is a braided diagram group.