Coarse median spaces and groups.
The author introduces and systematically studies `coarse median spaces'. These are metric spaces that admit coarse median structure that can be viewed as a coarse nonpositive curvature condition. The existence of such a median on a geodesic space is quasi-isometry invariant, and so it applies to finitely generated groups via their Cayley graphs. Many naturally occurring spaces and groups admit such structures and the author suggests that it is conceivable that every CAT(0) space does. Following the abstract, the contents of the paper can be briefly described as follows: ``We show that asymptotic cones of coarse median spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded Euclidean plane in the space. Using the centroid construction of \textit{J. A. Behrstock} and \textit{Y. N. Minsky} [J. Lond. Math. Soc., II. Ser. 84, No. 3, 765-784 (2011; Zbl 1269.20032)], we show that the mapping class group has this property, and recover the rank theorem of \textit{J. A. Behrstock} and \textit{Y. N. Minsky} [Ann. Math. (2) 167, No. 3, 1055-1077 (2008; Zbl 1280.57015)] and of \textit{U. Hamenstädt} [preprint, \url{arXiv:math/0512429}]. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.''
scientific article; zbMATH DE number 6168037