An isoperimetric function for a finitely presented group G bounds the number of defining relations needed to prove that a word w =G 1 in terms of the length of w. Suppose G = F/R is a finitely presented group where F is a finitely generated free group and R is the normal closure of a finite set of relators. For each w ∈ R we define the area ∆(w) to be the least number of uses of the given relat...

Coalgebras of set functors preserving weak pullbacks are particularly well behaved Invoking a result by Carboni Kelly and Wood we show that this can be explained by the fact that such functors can be uniquely extended to a relator This insight next suggests a de nition of metric bisimulation

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator g...

This paper investigates and clarifies some connections between the theory of one-relator groups special one-relation inverse monoids, i.e. those monoids with a presentation form [Formula: see text]. We show that every group admits monoid presentation. subsequently consider classes text] which can be defined by presentations in defining word is arbitrary; reduced; cyclically or positive, respect...

In this paper it is proved that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the two Dehn functions, of the group and of the manifold, respectively, are equivalent. 1. Dehn functions and their equivalence Let X be a simply connected 2-complex , and let w be an edge circuit in X. If D is a van Kampen di...