**Question.** Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree.

Things I know related to this: a minimal action on an $\mathbb{R}$-tree $T$ (like a simplicial tree) gives a canonical measured foliation $F$ on $S$, and a measured foliation in turn gives an action on an $\mathbb{R}$-tree $T_F$, with a surjective equivariant map $T_F \to T$. For simplicial trees, the measured foliation will come by taking a finite collection of simple closed curves and expanding them to make a measured foliation. The resulting tree $T_F$ is then an infinite-valence simplial tree, on which $\pi_1(S)$ acts faithfully. So the question is basically whether you can fold that infinite-valence tree to get a finite-valence tree on which the action is still faithful.

Another point of view is to look at the resulting graph of groups (from Bass-Serre theory). But the edge groups cannot be cyclic (Skora proved this), and it's not clear to me how to proceed from this point.

*Richard K. Skora*, **Splittings of surfaces**, *J. Amer. Math. Soc.* **9** (1996), no. 2, 605--616.