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

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

2012-04-07

arxiv.org/abs/1204.1613v1

Mathematics

Group Theory

18 pages, 5 figures

Scientific paper

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 1/n 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^{1/2} for general non-abelian nilpotent groups. As a corollary we also get an error term of the form vol(B(n))=cn^d + 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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