We introduce the notion of a
coarse median on a metric space. This satisfies the axioms of a median algebra up to
bounded distance. 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. We show that asymptotic cones of such 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 Behrstock and Minsky, we show that the mapping
class group has this property, and recover the rank theorem of Behrstock
and Minsky and of Hamenstädt. We explore various other properties of
such spaces, and develop some of the background material regarding median
algebras.
Keywords
median algebra, cube complex, rank, mapping class group
