Dehn Function - Generalizations

Generalizations

• There are several companion notions closely related to the notion of an isoperimetric function. Thus an isodiametric function bounds the smallest diameter (with respect to the simplicial metric where every edge has length one) of a van Kampen diagram for a particular relation w in terms of the length of w. A filling length function the smallest filling length of a van Kampen diagram for a particular relation w in terms of the length of w. Here the filling length of a diagram is the mimimum, over all combinatorial null-homotopies of the diagram, of the maximal length of intermediate loops bounding intermediate diagrams along such null-homotopies. The filling length function is closely related to the non-deterministic space complexity of the word problem for finitely presented groups. There are several general inequalities connecting the Dehn function, the optimal isodiametric function and the optimal filling length function, but the precise relationship between them is not yet understood.

• There are also higher-dimensional generalizations of isoperimetric and Dehn functions. For k ≥ 1 the k-dimensional isoperimetric function of a group bounds the minimal combinatorial volume of (k + 1)-dimensional ball-fillings of k-spheres mapped into a k-connected space on which the group acts properly and cocompactly; the bound is given as a function of the combinatorial volume of the k-sphere. The standard notion of an isoperimetric function corresponds to the case k = 1. Unlike the case of standard Dehn functions, little is known about possible growth types of k-dimensional isoperimetric functions of finitely presented groups for k ≥ 2.

• In his monograph Asymptotic invariants of infinite groups Gromov proposed a probabilistic or averaged version of Dehn function and suggested that for many groups averaged Dehn functions should have strictly slower asymptotics than the standard Dehn functions. More precise treatments of the notion of an averaged Dehn function or mean Dehn function were given later by other researchers who also proved that indeed averaged Dehn functions are subasymptotic to standard Dehn functions in a number of cases (such as nilpotent and abelian groups).

• A relative version of the notion of an isoperimetric function plays a central role in Osin' approach to relatively hyperbolic groups.

• Grigorchuk and Ivanov explored several natural generalizations of Dehn function for group presentations on finitely many generators but with infinitely many defining relations.