# Isoperimetric inequalities of a group

How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being $\delta-$hyperbolic in the sense of Gromov is expressible via Riemann integration?

In other words, how do you define "being $\delta-$hyperbolic group" by using integrals in metric groups?

(Note: I am not interested in the "Riemann" part, so you are free to take commutative groups with lebesgue integration etc.)

I should add that I actually prefer to prove that a linear (or subquadratic) isoperimetric inequality implies $\delta$-hyperbolicity using a coarse notion of area (see e.g. Bridson-Haefliger's book) or using Dehn functions, the latter can be found in Bridson's beautiful paper The geometry of the word problem.
• The last paragraph of t.b.'s answer can be interpreted as a form of integration of functions $f:R\to X$ where $X$ is the Cayley 2-complex of a finitely presented group $G$. This is just a fancy way of saying that you compute the area of a closed curve in the Cayley graph by adding up the number of 2-cells of a disc map into the Cayley 2-complex spanned by that curve, and minimizing over all such disc maps. After all, integration is just summation. – Lee Mosher May 22 '12 at 12:56