Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

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 oneended hyperbolic group contain a subgroup which is isomorphic to the fundamental group of a closed surface of genus at least 2? In [10] and [3] the authors constructed and classified all 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 KacMoody buildings that are not right-angled. Here we study surface subgroups of the 23 torsion free groups obtained in [10]. Generalizing a result by Constantine, Lafont and Oppenheim [4] to any cocompact action on hyperbolic buildings we see that surface subgroups exist in these groups, too. However, in most of the cases there are no periodic apartments invariant under the action of a genus two surface. The proof of the latter result involves computer search.