Coarse length can be unbounded in 3-step nilpotent Lie groups

Abstract

In "On the asymptotics of the growth of 2-step nilpotent groups" (J. London Math. Soc. (2), 58 (1998)), we remarked that, contrary to 2-step nilpotent simply connected Lie groups, in 3-step nilpotent simply connected Lie groups it is possible that `$\mathbb{R}$-words' in the given generators cannot be replaced by an equally long $\mathbb{R}$-word representing the same group element and having a bounded number of direction changes. In this note, we present an example for this phenomenon.