Isoperimetric functions of nilpotent Lie groups

等周函数描述了度量空间的子集的体积与其边界的表面积之间的关系。由填充函数构成了一类特殊的填充函数。它们测量了Lipschitz-k-圈的体积与填充这些圈的Lipschitz-(k+1)-链的体积之比。度量空间的填充函数的增长率是拟等距不变量，并解码重要的几何性质。例如，他们通过对称空间增长行为的变化来检测对称空间的秩.在这个项目中，我们集中在配备不变黎曼度量的幂零李群的填充函数上.格罗莫夫预测，它们的生长行为将发生类似于对称空间的变化。我们打算在证明这一猜想和阐明其几何意义方面取得进展。为此，我们研究了幂零李群的渐近圆锥和相应的次黎曼几何。

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.