On right-angled Artin groups whose underlying graphs have two vertices with the same link

Jongtae Kim; Myoungho Moon

doi:10.4134/BKMS.2013.50.2.543

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(2): 543-558

Printed March 31, 2013

https://doi.org/10.4134/BKMS.2013.50.2.543

On right-angled Artin groups whose underlying graphs have two vertices with the same link

Jongtae Kim and Myoungho Moon

KonKuk University, KonKuk University

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Abstract

Let $\Gamma$ be a graph which contains two vertices $a, b$ with the same link. For the case where the link has less than $3$ vertices, we prove that if the right-angled Artin group $A(\Gamma)$ contains a hyperbolic surface subgroup, then $A(\Gamma - \{a\})$ contains a hyperbolic surface subgroup. Moreover, we also show that the same result holds with certain restrictions for the case where the link has more than or equal to $3$ vertices.

Keywords: right-angled Artin group, hyperbolic surface subgroup

MSC numbers: Primary 20F36, 20F65; Secondary 05C25