Calculus of Variations and Geometric Measure Theory
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E. Breuillard - E. Le Donne

On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry

created by ledonne on 21 Dec 2018



Inserted: 21 dec 2018

Year: 2012

ArXiv: 1204.1613 PDF


Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1n towards its asymptotic cone. We show that due to the possible presence of abnormal geodesics in the asymptotic cone, this rate cannot be better than n{12} for general non-abelian nilpotent groups. As a corollary we also get an error term of the form vol(B(n))=cnd + O(n{d-2(3r)}) for the volume of Cayley balls of a nilpotent group with nilpotency class r. We also state a number of related conjectural statements.

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