**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**](http://dx.doi.org/10.1090/S0894-0347-96-00205-6), *J. Amer. Math. Soc.* **9** (1996), no. 2, 605--616.



Can a surface group act on a finite-valence simplicial tree?
Theorem 1.2 of Breuillard-Gelander-Souto-Storm in "Dense embeddings of surface groups" (<http://arxiv.org/abs/math/0602635>) says the following:


Let $G$ be a locally compact group. Suppose that $G$ contains a nondiscrete
free subgroup $F$ of finite rank $r > 1$. Then $G$ has a subgroup $\Gamma$ containing $F$ such that
$\Gamma$ is isomorphic to a surface group (of genus $2r$). In particular, if $G$ has a dense free subgroup of finite rank, then it has a dense surface group.


Apply this in the case $G$ being the automorphism group of a locally finite regular tree, which clearly contains a free group. Of course, you can always pass to a minimal sub-tree, or alternatively start with a dense free group and get a dense surface which thus will act minimally on the original tree.


