NILPOTENT GROUPS, ASYMPTOTIC CONES AND SUBFINSLER GEOMETRY
NILPOTENT GROUPS, ASYMPTOTIC CONES AND SUBFINSLER GEOMETRY
复制标题
幂函数群、渐进锥和次级几何
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
E. Donne
中科院分区：
文献类型：
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作者：
E. Breuillard;E. Donne
我们给出了一般有限生成幂零群的凯莱图向其渐近锥收敛的速度的估计。这会在大球体积的渐进中产生一个误差项，即 |B(n)| = cnd + O(nd−α)，其中误差项中的指数 α > 0 仅取决于幂零级。据推测，这对于 α = 1 成立。我们将此猜想与亚黎曼几何中的其他著名猜想联系起来，并表明异常测地线起着重要作用。我们还详细研究了海森堡群的几何结构（配备 Pansu 度量），并通过给出收敛到渐近锥的速度不快于 n− 1 2 的示例，表明我们的结果对于 2 步群来说是清晰的。
We give an estimate of the speed of convergence of Cayley graphs of general finitely generated nilpotent groups towards their asymptotic cone. This yields an error term in the asymptotics for the volume of large balls, namely |B(n)| = cnd + O(nd−α), where the exponent α > 0 in the error term depends only on the nilpotency class. Conjecturally this holds for α = 1. We relate this conjecture to other well-known conjectures in subRiemannian geometry and show that abnormal geodesics play an important role. We also study in some detail the geometry of the Heisenberg group (equipped with the Pansu metric) and show that our results are sharp for 2-step groups by giving an example for which the speed of convergence to the asymptotic cone is no faster than n− 1 2 .