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.
Highlights
In group theory, the study of intersections of subgroups has been recurrently considered in the literature
The strategy of the proof arises from the following crucial lemma given by Droms on the way of proving Theorem 1.2: Every nonempty Droms graph is either disconnected, or it contains a central vertex
Similar preserving properties concerning free and direct products were studied for the Membership Problem (MP) by K
Summary
The study of intersections of subgroups has been recurrently considered in the literature. In Zm Howson’s property is trivial, whereas for free groups it was proved by Howson himself in [15], where he gave an algorithm to compute generators for the intersection. Not far from these groups one can find examples without the Howson property: consider the group F2 × Z = a, b | − × t | − and the subgroups H = a, b and K = ta, b ; both. The goal of the present paper is to extend the algebraic arguments given there, in order to achieve similar properties for a much wider family of groups To this end it is convenient to consider the following variations for a general finitely presented group G. We note that finite {P4, C4}-free graphs have received diverse denominations throughout the literature, including comparability graphs of forests (in [29]), transitive forests (in [26]), trivially perfect graphs (in [4]), and quasi-threshold graphs (in [19])
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