Local homology properties of boundaries of groups.

*(English)*Zbl 0872.57005Let \(\widetilde X\) 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.

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.

Reviewer: G.E.Lang jun.(Fairfield)

##### MSC:

57M07 | Topological methods in group theory |