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 define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the G-CW-version EF(G) and the numerable G-space version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C ∗-algebra, for the Farrell-Jones Conjecture about the algebraic K-and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

Abstract. We show that the “hard ” part of Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly linear-time generic-case complexity. We also prove that the stabilizers of generic elements of Fk in Aut(Fk) are cyclic groups generated by inner automorphisms. We apply these results to one-relator groups and show that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We prove that the number In of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies c1 n (2k − 1)n ≤ In ≤ c2 n (2k − 1)n, where c1 = c1(k)> 0, c2 = c2(k)> 0 are some constants independent of n. Thus In grows in essentially the same manner as the number of cyclic words of length n.

this paper, we study 2-dimensional analogies of this idea: semigroup diagrams, monoid pictures, annular diagrams, cylindric pictures and braided pictures. While the groups of linear diagrams are all free, we get a large class of groups which are representable by 2-dimensional semigroup diagrams. Semigroup diagrams, are well-known geometrical objects used in the study of Thue systems (=semigroup presentations). They were first formally introduced by Kashintsev [16], see also Remmers [29], Stallings [34] or Higgins [13]. The role of semigroup diagrams in the study of semigroups is similar to the role of van Kampen diagrams in the study of groups (see [22] or [26])

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. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...

Abstract. We re-cast in a more combinatorial and computational form the topological approach of J.Stallings to the study of subgroups of free groups. 1.

Abstract. We survey current work relating to isoperimetric functions and isodiametric functions of finite presentations. §1. Introduction and Definitions Isoperimetric functions are classical in differential geometry, but their use in group theory derives from Gromov’s seminal article [Gr] and his characterization of word hyperbolic groups by a linear isoperimetric inequality. Isodiametric functions were introduced in our article [G1] in an attempt to provide a group theoretic

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In the eighth paper we use a modification of the sieve procedure, presented in [Se6] as part of the quantifier elimination procedure, to prove that free and torsion-free (Gromov) hyperbolic groups are stable. In the first 6 papers in the sequence on Diophantine geometry over groups we studied sets of solutions to systems of equations in a free group, and developed basic techniques and objects that are required for the analysis of sentences and elementary sets defined over a free group. The techniques we developed, enabled us to present an iterative procedure that analyzes EAE sets defined over a free group (i.e., sets defined using 3 quantifiers), and shows that every such set is in the Boolean algebra generated by AE sets ([Se6],41), hence, we obtained a quantifier elimination over a free group. In 1983 B. Poizat [Po1] proved that free groups are not super-stable (W. Hodges

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G�K � is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1- or 2-fellow traveler property. © 2000 Academic Press 1.