Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

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.

KEYWORDS:

• hyperbolic building
• surface subgroup
• periodic apartment

2000 AMS Subject Classification:

• Primary: 05E18
• Secondary: 20F67
• 51E24

Acknowledgments

The authors would like to thank Tim Steger for useful discussions. This article was finalized in July 2015 when the authors were invited to work at Max Planck Institute for Mathematics in Bonn. We would like to thank MPIM for their hospitality.

Funding

This work was supported by EPSRC grant EP/K016687/1.