  We show that several new classes of groups are measure strongly treeable. In
particular, finitely generated groups admitting planar Cayley graphs,
elementarily free groups, and the group of isometries of the hyperbolic plane
and all its closed subgroups. In higher dimensions, we also prove a dichotomy
that the fundamental group of a closed aspherical 3-manifold is either amenable
or has strong ergodic dimension 2. Our main technical tool is a method for
finding measurable treeings of Borel planar graphs by constructing one-ended
spanning subforests in their planar dual. Our techniques for constructing
one-ended spanning subforests also give a complete classification of the
locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.
