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.)   

You can do this using metric currents in the sense of Ambrosio-Kirchheim. This is a rather new development of geometric measure theory, triggered by Gromov and really worked out only in the last decade. I should warn you that this is rather technical stuff and nothing for the faint-hearted.
Urs Lang has a set of nice lecture notes, where you can find most of the relevant references, see here. 
My friend Stefan Wenger has done quite a bit of work on Gromov hyperbolic spaces and isoperimetric inequalities, his Inventiones paper Gromov hyperbolic spaces and the sharp isoperimetric constant seems most relevant. You can find a link to the published paper and his other work on his home page, the ArXiV-preprint is here.

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.
