Metabelian Groups and the Bieri-Groves Conjecture

Joseph Mullaney (U Glasgow)

Wednesday 22nd May, 2013 16:00-17:00 204


A group G is said to be metabelian if its derived group A is abelian. The Bieri-Strebel invariant ΣcA can be used to determine which finitely generated metabelian groups are finitely presented: this will be the case if and only if ΣcA does not contain any pair of antipodal points on the sphere Sn−1, where n is the rank of the finitely generated abelian group G/A. Bieri and Groves conjectured that ΣcA contains complete information about the higher cohomological finiteness conditions FPm satisfied by G, and not just finite presentability. We generalize an example first given by Baumslag of a metabelian group of 2 × 2 matrices over the field of rational functions in one variable, and consider the cohomological finiteness conditions these groups satisfy. We then explain how this relates to the Bieri-Groves conjecture. This is joint work with my PhD supervisor Peter Kropholler.

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