Extension properties of asymptotic property C and finite decomposition complexity. (English) Zbl 1400.20039
Asymptotic dimension was introduced by M. Gromov [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)] as a coarse analogue of covering dimension. In [Trans. Am. Math. Soc. 358, No. 11, 4749–4764 (2006; Zbl 1117.20032)], G. C. Bell and A. N. Dranishnikov proved an extension theorem for groups acting on metric spaces of finite asymptotic dimension by isometries.
In this paper, the authors introduce the notion of coarse quasi-action which is a generalization of actions by isometries, and prove extension theorems for the following coarse properties: finite asymptotic dimension (FAD); asymptotic property C (APC) introduced by A. N. Dranishnikov [Russ. Math. Surv. 55, No. 6, 1085–1129 (2000; Zbl 1028.54032); translation from Usp. Mat. Nauk 55, No. 6, 71–116 (2000)], finite decomposition complexity (FDC) introduced by E. Guentner et al., [Invent. Math. 189, No. 2, 315–357 (2012; Zbl 1257.57028)] and straight finite decomposition complexity (sFDC) introduced by A. Dranishnikov and M. Zarichnyi [Topology Appl. 169, 99–107 (2014; Zbl 1297.54064)].
The main theorem is the following:
Theorem. Let \(G\) be a finitely generated group with a coarse quasi-action on a metric space \(X\). The group \(G\) is equipped with a word metric. If \(X\) has FAD (respectively, APC) and the quasi-stabilizers have asymptotic dimension bounded from above by some number \(n \geq 0\), then \(G\) has FAD (respectively, APC). If \(X\) and all quasi-stabilizers have FDC (respectively, sFDC), then \(G\) has FDC (respectively, sFDC).
In this paper, the authors introduce the notion of coarse quasi-action which is a generalization of actions by isometries, and prove extension theorems for the following coarse properties: finite asymptotic dimension (FAD); asymptotic property C (APC) introduced by A. N. Dranishnikov [Russ. Math. Surv. 55, No. 6, 1085–1129 (2000; Zbl 1028.54032); translation from Usp. Mat. Nauk 55, No. 6, 71–116 (2000)], finite decomposition complexity (FDC) introduced by E. Guentner et al., [Invent. Math. 189, No. 2, 315–357 (2012; Zbl 1257.57028)] and straight finite decomposition complexity (sFDC) introduced by A. Dranishnikov and M. Zarichnyi [Topology Appl. 169, 99–107 (2014; Zbl 1297.54064)].
The main theorem is the following:
Theorem. Let \(G\) be a finitely generated group with a coarse quasi-action on a metric space \(X\). The group \(G\) is equipped with a word metric. If \(X\) has FAD (respectively, APC) and the quasi-stabilizers have asymptotic dimension bounded from above by some number \(n \geq 0\), then \(G\) has FAD (respectively, APC). If \(X\) and all quasi-stabilizers have FDC (respectively, sFDC), then \(G\) has FDC (respectively, sFDC).
Reviewer: Takamitsu Yamauchi (Matsuyama)
MSC:
| 20F69 | Asymptotic properties of groups |
| 20F65 | Geometric group theory |
| 20F05 | Generators, relations, and presentations of groups |
| 54F45 | Dimension theory in general topology |
Keywords:
asymptotic dimension; finite decomposition complexity; asymptotic property C; coarse quasi-action; extension theoremsReferences:
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