## Coarse Alexander duality and duality groups.(English)Zbl 1086.57019

Summary: The Scott-Shalen Core Theorem asserts that if the fundamental group $$G$$ of a 3-manifold $$M$$ is finitely generated then there is a compact codimension-$$0$$ submanifold $$N$$ for which the inclusion $$N\subseteq M$$ is a homotopy equivalence. Hence $$G$$ is finitely presentable, and it follows easily that all 3-manifold groups are coherent.
The main result of this paper is an algebraic analogue of the Core Theorem, for $$(n- 1)$$-dimensional duality groups acting freely on “coarse $$PD_n$$-spaces”. (This class of spaces includes the universal covering spaces of finite $$PD_n$$-complexes.) In particular, if $$G$$ is a one-ended $$FP_2$$ subgroup of infinite index in a $$PD_3$$-group then $$G$$ is the ambient group of a $$PD_3$$-group pair $$(G, \{H_i\})$$, and so $$G$$ contains surface subgroups $$H_i$$ . Many interesting constraints on the possible subgroups of $$PD_3$$-groups follow from this Algebraic Core Theorem. (Note however that it remains unknown whether $$PD_3$$-groups are coherent.) The main technique is an extension of Alexander duality to subcomplexes of coarse $$PD_n$$-complexes. The arguments are essentially homological, and the paper is self-contained. As there are $$PD_n$$-groups which are not finitely presentable and so which do not act freely and cocompactly on contractible simplicial complexes the argument is extended in the Appendix to cover such situations.

### MSC:

 57P10 Poincaré duality spaces 20J05 Homological methods in group theory

### Keywords:

Alexander duality; coarse topology; Core Theorem; $$PD_n$$-space
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