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Subgroups of direct products of limit groups

#### Martin R. Bridson, James Howie, Charles F. Miller III,
Hamish Short

#### Preprint, April 2007. Accepted by
the Annals of Mathematics, October 2007.

A subgroup of a direct product of $n$ limit groups

is of type $FP_n(\Q)$ if and only if it contains a subgroup

of finite index that is itself a product of (at most $n$)

limit groups. This settles a conjecture of Sela.