Thursday, October 30
Martin Bridson, University of Oxford
A theorem of Baumslag and Roseblade says that the only finitely presented subgroups of a direct product of free groups are the "obvious" ones; this was extended by Howie and I to all limit groups (in the sense of Sela). On the other hand, the finitely generated subgroups of direct products of free groups can be remarkably wild, as I shall illustrate.
Celebrated examples of Stallings (1961) and Bieri (1976) show that the finitely presented subgroups are more complicated when there are 3 or more factors, but quite how complicated has remained somewhat mysterious. Motivation for pursuing this comes from Delzant and Gromov who showed that understanding which subgroups of a direct product of surface groups are finitely presented is an important step towards the problem of determining which groups are fundamental groups of compact Kahler manifolds (eg complex projective varieties). And following Sela's fundamental work, it is natural to cast these problems in the context of (fully) residually free groups.
I shall discuss the recent progress that Howie, Miller, Short and I have made towards understanding these groups. This progress includes solutions to decision problems, new classes of examples, and a general criterion for determining when subgroups of general direct products are finitely presented and when they are of type FPn.
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