Asymptotic hereditary asphericity of metric spaces of asymptotic dimension 1.
\textit{T. Januszkiewicz} and \textit{J. Świątkowski} introduced the concept of asymptotic hereditary asphericity (AHA) [in Geom. Topol. 11, 727-758 (2007; Zbl 1188.20043)]. Using this property, they showed that there exist hyperbolic groups of arbitrary cohomological dimension containing no subgroups isomorphic to fundamental groups of closed nonpositively curved Riemannian manifolds of dimension greater than \(2\). AHA is a quasi-isometry invariant, so it is a group property for finitely generated groups with the word metric. It follows from [loc. cit.] that there exist AHA groups of arbitrary large asymptotic dimension. The result of the paper under review is that all metric spaces with asymptotic dimension \(1\) are AHA. In a preprint Osajda and Świątkowski show that many classes of groups are AHA, and conjecture that all asymptotically aspherical groups of asymptotic dimension \(2\) are AHA. This can be viewed as an analog of the Whitehead conjecture and the paper under review fully examines the case of dimension \(1\).
scientific article; zbMATH DE number 5835560
scientific article