In this talk we consider 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? The question remains unanswered, even though copious

partial results exist.

First we will briefly explain the idea of the construction of groups

acting simply transitively on the vertices of hyperbolic triangular

buildings with the minimal generalized quadrangle as the link. Then

we'll consider surface subgroups of the obtained 23 torsion free

groups. We will show, that in most of the groups there are no periodic

apartments invariant under an 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.

This talk is continuation to Alina Vdovina's talk on July 24th, but

can be followed also independently.