preprint
Inserted: 21 dec 2018
Year: 2012
Abstract:
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.