# F.g group with infinite ends not Q.I to a free group

Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group?

• If you restrict to torsion-free finitely generated groups: ($\infty$ many ends)$\Leftrightarrow$ (non-trivial free product), while (quasi-isometric to a free group) $\Leftrightarrow$ (free non-abelian).
The free product $\mathbb{Z}^2 \ast \mathbb{Z}$ has infinitely-many ends, but is not quasi-isometric to a free group: furthermore, it is not hyperbolic since it has $\mathbb{Z}^2$ as a subgroup.