Isoperimetric functions of nilpotent Lie groups

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Isoperimetric functions describe the relation between the volume of subsets of a metric space and the surface area of their boundaries. A special class of them is formed by the filling functions. These measure the ratio of the volume of Lipschitz-k-cycles and the volume of Lipschitz-(k +1)-chains filling those. The growth rate of the filling functions of a metric space are quasi-isometry invariants and decode important geometric properties. For instance, they detect the rank of a symmetric space by a change in their growing behaviour.In this project we concentrate on the filling functions of nilpotent Lie groups equipped with invariant Riemannian metrics. Gromov predicted a change of their growing behaviour similar to the one for symmetric spaces. We intend to make progress in proving this conjecture and clarify its geometric meaning. For this we examine the asymptotic cones of nilpotent Lie groups and the corresponding sub-Riemannian geometry.