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

TL;DR

This paper demonstrates that in 3-step nilpotent Lie groups, the length of certain group elements can be unbounded, unlike in 2-step groups where such lengths are bounded, highlighting a fundamental difference in their geometric structure.

Contribution

The paper provides the first explicit example showing unbounded length phenomena in 3-step nilpotent Lie groups, contrasting with known results for 2-step groups.

Findings

Unbounded length of words in 3-step nilpotent Lie groups

Difference between 2-step and 3-step nilpotent groups in word length behavior

Explicit example illustrating the phenomenon

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.