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Local homology properties of boundaries of groups. (English) Zbl 0872.57005
Let $$\widetilde X$$ be a Euclidean retract, ER (embeddable in some Euclidean space as a retract). A closed subset $$Z$$ of $$\widetilde X$$ is a $$Z$$-set if there is deformation $$h_t$$: $$\widetilde X\to \widetilde X$$ with $$h_0= \text{id}$$ and $$h_t(\widetilde X) \cap Z= \emptyset$$ for $$t>0$$ (two equivalent definitions are given). If $$G$$ is a group, a $$Z$$-structure on $$G$$ is a pair $$(\widetilde X,Z)$$ such that $$\widetilde X$$ is an ER, $$Z$$ a $$Z$$-set in $$\widetilde X$$, $$X= \widetilde X \smallsetminus Z$$ admits a covering space action of $$G$$ with compact quotient, and the collection of translates of a compact set in $$X$$ forms a null sequence in $$\widetilde X$$ (for every open cover $$U$$ of $$\widetilde X$$ all but finitely many translates are $$U$$-small). A space $$Z$$ is a boundary of $$G$$ if there is a $$Z$$-structure, $$(\widetilde X,Z)$$, on $$G$$. These boundaries, which are not unique, are the topic of this paper.
After providing a number of examples of $$Z$$-structures on groups, it is shown that the shape of a boundary of a group and its dimension over a PID are determined by the cohomology of the group. The local properties of boundaries of groups are given in two theorems. The first assumes that $${\mathbf L}$$ is a countable field and relates $$H_q(Z,Z \smallsetminus \{z\})$$ to the finite dimensionality of $$H^{q+1} (G; {\mathbf L} G)$$. In the second, let $$U$$ be any non-empty open set in $$Z$$. If $$H^{q+1} (G; {\mathbf L}G)$$ is finitely generated then $$H^q (Z,Z \smallsetminus U) \to H^q (Z)$$ is onto, if not the image in $$H^q(Z)$$ contains an isomorphic copy of every finitely generated submodule of $$H^{q+1} (G; {\mathbf L} G)$$. With $${\mathbf L}$$ a countable field and an additional hypothesis (“Axiom $$H$$”) which will hold if $$G$$ is word-hyperbolic and $$Z$$ is the Gromov boundary, it is shown that either $$H_q (Z)\to H_q (Z,Z \smallsetminus \{z\})$$ is an isomorphism of finite dimensional vector spaces or $$H_q (Z,Z \smallsetminus \{z\})$$ is uncountable.
The second section studies Poincaré duality groups, groups which act freely, properly discontinuously, and cocompactly on a contractible cell complex $$Y$$ with $$H^*_c (Y)= H_c^* (\mathbb{R}^n)$$. The induced action of $$G$$ on $$H_c^*(Y)$$ is the orientation character. The main theorem states that if $$G$$ is a Poincaré duality group of dimension $$n$$ over $${\mathbf L}$$ such that the image of its orientation character is finite and $$Z$$ is a boundary of $$G$$, then $$Z$$ is a homology $$n$$-manifold, in fact a homology $$(n-1)$$ sphere over $${\mathbf L}$$.
The paper closes with a number of open questions and a section noting analogies between group theory and the theory of finite-dimensional compact metrizable spaces.

##### MSC:
 57M07 Topological methods in group theory
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