LAUSR

Abstract 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? In [Kangaslampi and Vdovina 10] and [Carbone et al. 12] 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 Kac–Moody buildings that are not right-angled. Here we study surface subgroups of the 23 torsion-free groups obtained in [Kangaslampi and Vdovina 10]. 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 2 surface. The existence of such an action implies 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.