21.79 (2026)

Let $G$ be a finitely generated group with a fixed finite generating set $S$ and the corresponding word metric $L_S(*)$. An element $g$ is said to be distorted in $G$ if $L_S(g^n)/n \to 0$ as $n \to \infty$; this notion is independent of the choice of the generating set $S$. For any, not necessarily finitely generated, group $H$, an element $g \in H$ is said to be distorted if there is a finitely generated subgroup $G$ of $H$ containing $g$ in which $g$ is distorted. Do there exist finitely generated left-orderable groups in which every nontrivial element is distorted?

Note that it is straightforward to construct countable (not finitely generated) left orderable groups with this property using HNN-extensions and applying results of V. V. Bludov and A. M.W. Glass.