## Topology & Geometric Group Theory Seminar

## Fall 2008

### 1:30 – 2:30, Malott 406

Thursday, October 30

**Martin
Bridson**, University of Oxford

*Subgroups of direct products of surface, free and limit groups*

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 FP_{n}.

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