8 research outputs found

    Cohomological Finiteness Conditions in Bredon Cohomology

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    We show that any soluble group GG of type Bredon-\FP_{\infty} with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type \FP_{\infty} must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-\FP_n for some n≤3n \leq 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.Comment: Corrected a mistake in Lemma 2.4 of the previous version, which had an effect on the results in Section 5 (the condition that all centralisers of infinite order elements are of type FP∞FP_\infty was added

    A note on the Mittag–Leffler condition for Bredon-modules

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    In this note we show the Bredon-analogue of a result by Emmanouil and Talelli, which gives a criterion when the homological and cohomological dimensions of a countable group GG agree. We also present some applications to groups of Bredon-homological dimension 11.Comment: 10 page

    Cohomoloical finiteness conditions in Bredon cohomology, preprint 2008

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    Abstract. We show that soluble groups G of type Bredon-FP ∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies the group is polycyclic. Another important ingredient of the proof is that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-FPn for some n ≤ 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups. 1
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