The authors of this article believe there is or should be a research area appropriately referred to as computational topology. Its agenda includes the identification and formalization of topological questions in computer applications and the study of algorithms for topological problems. It is hoped this article can contribute to the creation of a computational branch of topology with a unifying influence on computing and computer applications. Keywords. Survey; topology, geometry, algorithms, computer applications. INTRODUCTION The title of this article combines computation with topology, suggesting a general research activity that studies the computational aspects of problems with topological flavor. What we have in mind is distinctly different from studying the topology of computing or the computer animation of topology. Computational studies of topological questions can be found in the mathematics and the computer science literature, but no concerted effort is apparent. The auth...

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is

The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...

This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995

. It is shown that, for site percolation on the Cayley graph of a co-compact Fuchsian group of genus 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = Pfsite is openg. In such cases, the set of limit points at 1 of an infinite cluster is shown to be a perfect, nowhere dense set of Lebesgue measure 0. These results are also shown to hold for a class of hyperbolic triangle groups. 1. Introduction Percolation on a "Euclidean" graph, such as the standard integer lattice Z d , exhibits a single threshold probability p c , above which infinite clusters exist with probability 1 and below which they exist with probability 0. In the percolation regime p ? p c the infinite cluster is unique [2]. The purpose of this paper is to show that percolation on a "noneuclidean" graph may exhibit several threshold probabilities, and in particular that for some values of p infinitely many infinite clusters may coexist, while for other values of p...

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automaton subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership. © 2006 Elsevier Inc. All rights reserved.

Abstract. Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [12]. Using the definition of Farb of a relatively hyperbolic group [9], we prove this assertion. We conclude that the conjugacy problem is solvable for the following two classes of groups: fundamental groups of complete, finite-volume, negatively curved manifolds, and finitely generated fully residually free groups.