# Hyperbolic Triangular Buildings without Periodic Planes of Genus 2

Research output: Contribution to journal › Article › Scientific › peer-review

### Standard

**Hyperbolic Triangular Buildings without Periodic Planes of Genus 2.** / Kangaslampi, Riikka; Vdovina, Alina.

Research output: Contribution to journal › Article › Scientific › peer-review

### Harvard

*Experimental Mathematics*, vol. 26, no. 1, pp. 54-61. https://doi.org/10.1080/10586458.2015.1110541

### APA

*Experimental Mathematics*,

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

### Vancouver

### Author

### Bibtex - Download

}

### RIS - Download

TY - JOUR

T1 - Hyperbolic Triangular Buildings without Periodic Planes of Genus 2

AU - Kangaslampi, Riikka

AU - Vdovina, Alina

PY - 2017

Y1 - 2017

N2 - 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 [Kangaslampiand 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.

AB - 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 [Kangaslampiand 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.

KW - hyperbolic building

KW - surface subgroup

KW - periodic apartment

U2 - 10.1080/10586458.2015.1110541

DO - 10.1080/10586458.2015.1110541

M3 - Article

VL - 26

SP - 54

EP - 61

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 1

ER -

ID: 6484575