Hyperbolic Triangular Buildings without Periodic Planes of Genus 2

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

In: Experimental Mathematics, Vol. 26, No. 1, 2017, p. 54-61.

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Kangaslampi, Riikka ; Vdovina, Alina. / Hyperbolic Triangular Buildings without Periodic Planes of Genus 2. In: Experimental Mathematics. 2017 ; Vol. 26, No. 1. pp. 54-61.

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@article{b40b5096e3294a6e9aa2d477feb0f7bc,
title = "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 [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.",
keywords = "hyperbolic building, surface subgroup, periodic apartment",
author = "Riikka Kangaslampi and Alina Vdovina",
year = "2017",
doi = "10.1080/10586458.2015.1110541",
language = "English",
volume = "26",
pages = "54--61",
journal = "Experimental Mathematics",
issn = "1058-6458",
number = "1",

}

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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