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Hyperbolic Triangular Buildings Without Periodic Planes of Genus 2

Lookup NU author(s): Dr Alina Vdovina

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).

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 etal. 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.

Publication metadata

Author(s): Kangaslampi R, Vdovina A

Publication type: Article

Publication status: Published

Journal: Experimental Mathematics

Year: 2017

Volume: 26

Issue: 1

Pages: 54-61

Print publication date: 01/01/2017

Online publication date: 13/07/2016

Acceptance date: 16/10/2015

Date deposited: 01/06/2017

ISSN (print): 1058-6458

ISSN (electronic): 1944-950X

Publisher: Taylor & Francis Inc.

URL: http://dx.doi.org/10.1080/10586458.2015.1110541

DOI: 10.1080/10586458.2015.1110541

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