Co-contractions of graphs and right-angled Artin groups

Abstract

We define an operation on finite graphs, called co-contraction. Then we show that for any co-contraction (Gamma) over cap of a finite graph Gamma, the right-angled Artin group on Gamma contains a subgroup which is isomorphic to the right-angled Artin group on (Gamma) over cap. As a corollary, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the right-angled Artin groups on those graphs contain hyperbolic surface groups. This gives the negative answer to a question raised by Gordon, Long and Reid.