多项式增长的局部紧群几何与大球的形状
Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls
摘要：
得到了任意局部紧群G中大球体积的渐近性质。这是通过研究G的几何和P.Pansu的论文的推广来完成的。特别地，我们证明了任何这样的G都弱可分解到某个单连通可解李群S，即G的Lie影子。我们还证明了G中的大球具有渐近形状，即经过适当的重正化后，它们收敛到一个极限紧集，这个极限紧集可以用几何方法来解释。然后我们讨论了收敛速度，处理一些例子，并给出了应用遍历论。我们还回答了Burago关于左不变度量的一个问题，并恢复了Stoll关于幂零群增长序列的非理性的一些结果.
英文摘要：
We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.
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