Equivalence of Geometric and Combinatorial Dehn Functions

TL;DR

This paper proves that for certain groups acting on simply connected Riemannian manifolds, the geometric and combinatorial Dehn functions are equivalent, linking geometric and algebraic complexity measures.

Contribution

It establishes the equivalence of geometric and combinatorial Dehn functions for groups acting properly discontinuously and cocompactly on simply connected Riemannian manifolds.

Findings

Geometric and combinatorial Dehn functions are equivalent under specified conditions.

The result connects geometric and algebraic properties of groups.

Provides a bridge between geometric group theory and Riemannian geometry.

Abstract

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 the manifold, respectively, are equivalent.