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Enriched pullbacks and the intersection problem

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The classic theory of Stallings describing subgroups of free groups as automata provides a very convenient (and fully algorithmic) way of understanding the intersection of two finitely generated subgroups H1,H2 of Fn. Namely, the automata associated to the intersection is precisely the core of the connected component containing the basepoint of the pullback of the automata associated to H1 and H2. This scenario immediately provides a finite basis for the intersection (and thus Howson's property for free groups), as well as the classic Hanna Neumann bound for the rank of such an intersection. We extend the former description to subgroups of Zm x Fn by admitting abelian labels in the edges and modifying consequently the folding process. This allows us to describe the automata associated to the intersection of two finitely generated subgroups as the one obtained from the enriched pullback of the free projections of the intersecting subgroups by "exploding" the edges according to the abelian labels it has. Since these "explosions" can be infinite, we immediately obtain that (nondegenerate) free-abelian times free groups are not Howson. On the other side, when all the explosions are finite, it becomes clear that they can be finite of unbounded size. Thus, (even for finitely generated intersections) no "Hanna Neumann"-like inequality can be expected for any family containing nondegenerated free-abelian times free-groups. This is joint work with Enric Ventura.

Enric Ventura

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Year of publication: 2019

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