Intersection problem for Droms RAAGs

Abstract

We solve the subgroup intersection problem (SIP) for any RAAG [Formula: see text] of Droms type (i.e. with defining graph not containing induced squares or paths of length [Formula: see text]): there is an algorithm which, given finite sets of generators for two subgroups [Formula: see text], decides whether [Formula: see text] is finitely generated or not, and, in the affirmative case, it computes a set of generators for [Formula: see text]. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. [Formula: see text]) even have unsolvable SIP.