We prove that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the Dehn function of the group and the corresponding filling function of the manifold are equivalent, in a sense described below. 1. Dehn functions and their equivalence Let X be a simply connected 2-complex , and let w be an edge circuit in X....

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A ∗C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

The focus of this article is on two forms of isometries and homomorphisms in proximal relator spaces. A practical outcome of this study is the detection of descriptively near, disjoint sets in proximity spaces with application in the study of proximal algebraic structures.

We give examples of CAT(0), biautomatic, free–by–cyclic, one–relator groups which have finite–rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules–versus–the–hydra battles in string– rewriting. 2010 Mathematics Subject Classification: 20F65, 20F10...

Processes can be seen as relations extended in time. In this paper we want to investigate this observation by deriving an ordered category of processes. We model processes as co-algebras of a relator on Dedekind category up to bisimilarity. On those equivalence classes we define a lower semi-lattice structure and a monotone composition operation.

We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE. The main technical result of the paper states that, for certain finit...

The classical results of Magnus on one-relator groups (such as the Freiheitssatz and the solvability of the generalized word problem for Magnus subgroups) are generalized to some (nitely presented groups whose relators are constructed inductively. The diagrammatic asphericity of such group presentations is also proven. c © 2001 Elsevier Science B.V. All rights reserved. MSC: Primary 20F05; 20F0...

Exponential equations in free groups were studied initially by Lyndon and Schützenberger [26] and then by Comerford and Edmunds [5]. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential equations have solution is decidable, in finitely generated free groups. In this paper we show that for finite systems of quadratic exponential equations...