  It is well known that surface groups admit free and proper actions on finite
products of infinite valence trees. In this note, we address the question of
whether there can be a free and proper action on a finite product of bounded
valence trees. We provide some obstructions and an arithmetic criterion for
existence. The bulk of the paper is devoted to an approach to verifying the
arithmetic criterion by studying the character variety of certain surface
groups over fields of positive characteristic. The methods may be useful for
attempting to determine when groups admit good linear representations in other
contexts.
