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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.)

user1729
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niyazi
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1 Answers1

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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.

t.b.
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  • Thank you Theo, I need some time to comprehend your message. – niyazi Mar 18 '11 at 05:19
  • @niyazi: It is more a list of references addressing the second version of your question (and some variants) than an answer, but I think that this is the closest you can get. – t.b. Mar 18 '11 at 05:23
  • 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