We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial ...

The interesting subsets of a Peano space which were later called the arc-sets or A -sets were defined independently and in a different manner by G. T. Whyburn and the author, and each studied the properties of these sets which have proved so useful in the later theory. The similarity of the properties led shortly to the conjecture that a relation existed between the two definitions, and investi...

Using space-filling curves to order multidimensional data has been found to be useful in a variety of application domains. This paper examines the space-filling curve induced ordering of multidimensional data that has been transformed using shape preserving transformations. It is demonstrated that, although the orderings are not invariant under these transformations, the probability of an order...

We prove that many features of Thurston’s Dehn surgery theory for hyperbolic 3manifolds generalize to Einstein metrics in any dimension. In particular, this gives large, infinite families of new Einstein metrics on compact manifolds.

Hilbert's spacelling curve is a continuous function that maps the unit interval onto the unit square. The construction of such curves in the 1890s surprised mathematicians of the time and led, in part, to the development of dimension theory. In this paper, we discuss how modern notions of selfsimilarity illuminate the structure of this curve. In particular, we show that Hilbert's curve has a ba...

Hilbert’s two-dimensional space-filling curve is appreciated for its good locality properties for many applications. However, it is not clear what is the best way to generalize this curve to filling higher-dimensional spaces. We argue that the properties that make Hilbert’s curve unique in two dimensions, are shared by 10 694 807 structurally different space-filling curves in three dimensions. ...

Hilbert curves can be generated by using L-System concept. A theoretical intuitive method is suggested for manipulating the Hilbert space filling curve for simple coding to one dimension. Our method is aimed to code the Hilbert curve simply by observing the base Hilbert curve as generated. By simple observation one can code the Hilbert curve easily. The present work has been mainly inspired by ...