Weak \(\mathcal{Z}\)-structures and one-relator groups. (English) Zbl 1542.57014
Motivated by the notion of boundary for hyperbolic and \(CAT(0)\) groups, M. Bestvina and M. Feighn [J. Differ. Geom. 43, No. 4, 783–788 (1996; Zbl 0862.57027)] introduced the notion of a (weak) \(\mathcal{Z}\)-structure and (weak) \(\mathcal{Z}\)-boundary for a group \(G\) of type \(\mathcal{F}\) (i.e., having a finite \(K(G,1)\) complex), with some implications on the Novikov conjecture for \(G\).
A \(\mathcal{Z}\)-structure on a group \(G\) is a pair \((W, Z)\) of spaces satisfying the following conditions:
(1) \(W\) is a contractible \(ANR\),
(2) \(Z\) is a \(\mathcal{Z}\)-set in \(W\),
(3) \(X = W - Z\) admits a proper, free and cocompact action by \(G\), and
(4) For any open cover of \(W\), and any compact subset \(K \subseteq X\), all but finitely many translates of \(K\) lie in some element of the cover (nullity condition).
If only Conditions (1)-(3) are satisfied, then \((W, Z)\) is called a weak \(\mathcal{Z}\)-structure on \(G\), and \(Z\) is a (weak) \(\mathcal{Z}\)-boundary for \(G\).
It is known that a group satisfying (1)–(4) must be of type \(\mathcal{F}\). The question whether or not every group of type \(\mathcal{F}\) admits a weak \(\mathcal{Z}\)-structure remains open (for some classes of groups it is shown for example in [C. R. Guilbault, Algebr. Geom. Topol. 14, No. 2, 1123–1152 (2014; Zbl 1316.57002)]).
In this paper the authors prove that every torsion free one-relator group admits a weak \(\mathcal{Z}\)-structure; and, in the 1-ended case the corresponding weak \(\mathcal{Z}\)-boundary has the shape of either a circle or a Hawaiian earring depending on whether the group is a virtually surface group or not (Theorem 1.2). They also extend this result to some wider class of groups called \(\mathcal{C}\) satisfying a Freiheitssatz property.
Theorem 1.6. Every torsion free, 1-ended generalized one-relator group \(G \in \mathcal{C}\) admits a weak \(\mathcal{Z}\)-structure. Moreover, the corresponding weak \(\mathcal{Z}\)-boundary has the shape of either a circle or a Hawaiian earring depending on whether \(G\) is a virtually surface group or not.
A \(\mathcal{Z}\)-structure on a group \(G\) is a pair \((W, Z)\) of spaces satisfying the following conditions:
(1) \(W\) is a contractible \(ANR\),
(2) \(Z\) is a \(\mathcal{Z}\)-set in \(W\),
(3) \(X = W - Z\) admits a proper, free and cocompact action by \(G\), and
(4) For any open cover of \(W\), and any compact subset \(K \subseteq X\), all but finitely many translates of \(K\) lie in some element of the cover (nullity condition).
If only Conditions (1)-(3) are satisfied, then \((W, Z)\) is called a weak \(\mathcal{Z}\)-structure on \(G\), and \(Z\) is a (weak) \(\mathcal{Z}\)-boundary for \(G\).
It is known that a group satisfying (1)–(4) must be of type \(\mathcal{F}\). The question whether or not every group of type \(\mathcal{F}\) admits a weak \(\mathcal{Z}\)-structure remains open (for some classes of groups it is shown for example in [C. R. Guilbault, Algebr. Geom. Topol. 14, No. 2, 1123–1152 (2014; Zbl 1316.57002)]).
In this paper the authors prove that every torsion free one-relator group admits a weak \(\mathcal{Z}\)-structure; and, in the 1-ended case the corresponding weak \(\mathcal{Z}\)-boundary has the shape of either a circle or a Hawaiian earring depending on whether the group is a virtually surface group or not (Theorem 1.2). They also extend this result to some wider class of groups called \(\mathcal{C}\) satisfying a Freiheitssatz property.
Theorem 1.6. Every torsion free, 1-ended generalized one-relator group \(G \in \mathcal{C}\) admits a weak \(\mathcal{Z}\)-structure. Moreover, the corresponding weak \(\mathcal{Z}\)-boundary has the shape of either a circle or a Hawaiian earring depending on whether \(G\) is a virtually surface group or not.
Reviewer: Danuta Kołodziejczyk (Warszawa)
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