Weak \(\mathcal{Z}\)-structures for some classes of groups. (English) Zbl 1316.57002
The paper under review is very interesting, useful and well written. It studies some “nice” compactifications of spaces on which a given group acts “nicely”.
M. Bestvina in [Mich. Math. J. 43, No. 1, 123–139 (1996; Zbl 0872.57005)] tried to formalize the concept of boundary of a group, defining the notion of \(\mathcal Z\)-structures and \(\mathcal Z\)-boundaries on groups. More precisely, on a given group \(G\) such a \(\mathcal Z\)-structure is defined as a pair of spaces \((\tilde X, Z)\), where \(\tilde X\) is a Euclidean retract (namely a compact, metrizable, finite-dimensional, contractible and locally-contractible space), and \(Z\) is a \(Z\)-set in \(\tilde X\) (namely a closed subset such that, for every open \(U \subset \tilde X\), the inclusion \(U - Z \hookrightarrow U\) is a homotopy equivalence), satisfying the following two conditions: {} (a) \(\tilde X - Z\) admits a co-compact covering space action of \(G\); {} (b) for every open cover \(\mathcal U\) of \(\tilde X\), and for any compact \(K \subset \tilde X\), all but finitely many \(G\)-translates of \(K\) are contained in some element \(U\) of \(\mathcal U\).
If the pair \((\tilde X, Z)\) satisfies only condition \((a)\), then it is called a weak \(\mathcal Z\)-structure on \(G\) and \(Z\) is a weak \(\mathcal Z\)-boundary of \(G\) (note that they are not unique).
The main class of groups admitting \(\mathcal Z\)-structures is the class of torsion-free word-hyperbolic groups, and M. Bestvina has even asked whether every type \(F\) group admits a \(\mathcal Z\)-structure or at least a weak \(\mathcal Z\)-structure.
The paper under review deals with this question, and the main theorems prove existence results of weak \(\mathcal Z\)-structures for several classes of groups.
More precisely, it is shown that the following classes of groups admit a weak \(\mathcal Z\)-structure: {} - extensions of nontrivial type \(F\) groups by nontrivial type \(F\) groups; {} - groups of type \(F\) with nontrivial center; {} - one-ended groups of type \(F\) with pro-monomorphic fundamental groups at infinity (and then, in particular, groups that are simply connected at infinity).
The techniques employed in the proofs make use of geometric group theory, topology at infinity, group actions and proper homotopy theory.
M. Bestvina in [Mich. Math. J. 43, No. 1, 123–139 (1996; Zbl 0872.57005)] tried to formalize the concept of boundary of a group, defining the notion of \(\mathcal Z\)-structures and \(\mathcal Z\)-boundaries on groups. More precisely, on a given group \(G\) such a \(\mathcal Z\)-structure is defined as a pair of spaces \((\tilde X, Z)\), where \(\tilde X\) is a Euclidean retract (namely a compact, metrizable, finite-dimensional, contractible and locally-contractible space), and \(Z\) is a \(Z\)-set in \(\tilde X\) (namely a closed subset such that, for every open \(U \subset \tilde X\), the inclusion \(U - Z \hookrightarrow U\) is a homotopy equivalence), satisfying the following two conditions: {} (a) \(\tilde X - Z\) admits a co-compact covering space action of \(G\); {} (b) for every open cover \(\mathcal U\) of \(\tilde X\), and for any compact \(K \subset \tilde X\), all but finitely many \(G\)-translates of \(K\) are contained in some element \(U\) of \(\mathcal U\).
If the pair \((\tilde X, Z)\) satisfies only condition \((a)\), then it is called a weak \(\mathcal Z\)-structure on \(G\) and \(Z\) is a weak \(\mathcal Z\)-boundary of \(G\) (note that they are not unique).
The main class of groups admitting \(\mathcal Z\)-structures is the class of torsion-free word-hyperbolic groups, and M. Bestvina has even asked whether every type \(F\) group admits a \(\mathcal Z\)-structure or at least a weak \(\mathcal Z\)-structure.
The paper under review deals with this question, and the main theorems prove existence results of weak \(\mathcal Z\)-structures for several classes of groups.
More precisely, it is shown that the following classes of groups admit a weak \(\mathcal Z\)-structure: {} - extensions of nontrivial type \(F\) groups by nontrivial type \(F\) groups; {} - groups of type \(F\) with nontrivial center; {} - one-ended groups of type \(F\) with pro-monomorphic fundamental groups at infinity (and then, in particular, groups that are simply connected at infinity).
The techniques employed in the proofs make use of geometric group theory, topology at infinity, group actions and proper homotopy theory.
Reviewer: Daniele Ettore Otera (Vilnius)
MSC:
| 57M07 | Topological methods in group theory |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 20F65 | Geometric group theory |
| 57N20 | Topology of infinite-dimensional manifolds |
Keywords:
type \(F\) groups; \(\mathcal Z\)-structures; \(\mathcal Z\)-compactification; proper homotopy equivalence; simple connectivity at infinity; group actionsCitations:
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