The large scale geometry of nilpotent Lie groups

Details The large scale geometry of nilpotent Lie groups The large scale geometry of nilpotent Lie groups

1999-03-15

arxiv.org/abs/math/9903089v2

Comm. Anal. Geom. 5(5) pp 951-982, 2001

Mathematics

Differential Geometry

22 pages, 1 figure. The revision corrects several typographical errors and makes the notation concerning metric spaces of gene

Scientific paper

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Caratheodory metrics and complete metric spaces.

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