Distortion of distances in nilpotent Lie groups (Q1416509)

Let \(G\) be a connected nilpotent Lie group and \(e\) its unit element. Let \(\Omega\) be a compact symmetric neighborhood of \(e\), and \(\Omega= \{e\}\), \(\Omega^1= \Omega,\dots, \Omega^n= \Omega\cdot\Omega\cdots\Omega\) (\(n\) times). For \(g\in G\), put \(| g|_G=| g|= \inf\{n,g\in\Omega^n\}\) and \[ d(g,h)= d(e,g^{-1}h)=| g^{-1} h|_G\quad (g, h\in G). \] This defines on \(G\) a left invariant distance, and if we take another \(\Omega'\), then we get a new distance equivalent to \(d\). The author first gives an estimate of \(| g|_G\) in terms of the canonical coordinates of second kind in \(G\). He next considers a connected subgroup \(H\) of \(G\) and, using his first result, proves that \(| h|_H\) \((h\in H)\) is bounded by a polynomial function of \(| h|_G\). The degree of this polynomial turns out to be the nilpotency rank of \(G\). When \(H\) is a one-parameter subgroup of \(G\), the main results of this paper had been announced by \textit{M. Gromov} [Asymptotic invariants of infinite groups, Geometric Group Theory, Vol. 2 (London Mathematical Society, Lecture Note Series 182, University Press, Cambridge) (1993; Zbl 0841.20039)].