### 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 sub-

group implies the existence of a periodic apartment. These groups are the first candidates for groups that have no surface subgroups arising from periodic apartments.

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 sub-

group implies the existence of a periodic apartment. These groups are the first candidates for groups that have no surface subgroups arising from periodic apartments.

Original language | English |
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Pages (from-to) | 54-61 |

Journal | Experimental Mathematics |

Volume | 26 |

Issue number | 1 |

Early online date | 13 Jul 2016 |

DOIs | |

Publication status | Published - 2017 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- hyperbolic building
- surface subgroup
- periodic apartment

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## Cite this

Kangaslampi, R., & Vdovina, A. (2017). Hyperbolic Triangular Buildings without Periodic Planes of Genus 2.

*Experimental Mathematics*,*26*(1), 54-61. https://doi.org/10.1080/10586458.2015.1110541