Equivalence of Geometric and Combinatorial Dehn Functions

TL;DR

This paper establishes that for groups acting nicely on simply connected Riemannian manifolds, the algebraic Dehn function and the geometric filling function are essentially the same, linking group theory and geometry.

Contribution

It proves the equivalence of the algebraic Dehn function and the geometric filling function for groups acting properly discontinuously and cocompactly on simply connected Riemannian manifolds.

Findings

Dehn function and filling function are equivalent under specified conditions

Links algebraic and geometric properties of groups and manifolds

Provides a unified understanding of filling invariants in geometric group theory

Abstract

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.