Intersection problem for Droms RAAGs
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We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether HnK is finitely generated or not, and, in the affirmative case, it computes a set of generators for HnK. 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. F_2 x F_2) even have unsolvable SIP.
CitationDelgado, J., Ventura, E., Zakharov, A. Intersection problem for Droms RAAGs. "International journal of algebra and computation", 1 Gener 2018, vol. 28, núm. 7, p. 1129-1162.