  We study surface subgroups of groups acting simply transitively on vertex
sets of certain hyperbolic triangular buildings. The study is motivated by
Gromov's famous surface subgroup question: Does every one-ended hyperbolic
group contain a subgroup which is isomorphic to the fundamental group of a
closed surface of genus at least 2? Here we consider surface subgroups of the
23 torsion free groups acting simply transitively on the vertices of hyperbolic
triangular buildings of the smallest non-trivial thickness. These groups gave
the first examples of cocompact lattices acting simply transitively on vertices
of hyperbolic triangular Kac-Moody buildings that are not right-angled. With
the help of computer searches we show, that in most of the cases there are no
periodic apartments invariant under the action of a genus two surface. The
existence of such an action would imply the existence of a surface subgroup,
but it is not known, whether the existence of a surface subgroup implies the
existence of a periodic apartment. These groups are the first candidates for
groups that have no surface subgroups arising from periodic apartments.
