ON RIGHT-ANGLED ARTIN GROUPS WHOSE UNDERLYING GRAPHS HAVE TWO VERTICES WITH THE SAME LINK

Abstract

Abstract. Let Γ be a graph which contains two vertices a,b with thesame link. For the case where the link has less than 3 vertices, we provethat if the right-angled Artin group A(Γ) contains a hyperbolic surfacesubgroup, then A(Γ−{a}) contains a hyperbolic surface subgroup. More-over, we also show that the same result holds with certain restrictions forthe case where the link has more than or equal to 3 vertices. 1. IntroductionLet Γ be a finite simple graph with the vertex set V(Γ) = {v 1 ,...,v n }.Recall that a finite simple graph is a finite graph without loops and multipleedges. In a finite simple graph, for a pair of two vertices, there is at most oneedge having them as endpoints. A right-angled Artin group A(Γ) on the graphΓ is the group given by the presentation with generators v 1 ,...,v n and definingrelators [v i ,v j ] whenever there is an edge having v i and v j as endpoints. Thegraph Γ is called the defining graph of A(Γ). It is known that two right-angledArtin groups are isomorphic if and only if their defining graphs are isomorphicas graphs ([6]).A subgraph Γ