Short hierarchically hyperbolic groups I: uncountably many coarse median structures

Abstract: We prove that the mapping class group of a sphere with five punctures admits uncountably many non-equivalent, coarsely equivariant coarse median structures, falsifying a folklore belief. The same is shown for right-angled Artin groups whose defining graphs are connected, triangle- and square-free, and have at least three vertices. Remarkably, in the latter case the coarse medians we produce are not induced by cocompact cubulations. In the process, we develop the theory of short hierarchically hyperbolic groups (HHGs), which also include Artin groups of large and hyperbolic type, graph manifolds groups, and extensions of Veech groups. We develop tools to modify their hierarchical structure, including using quasimorphisms to construct quasilines that serve as coordinate spaces, and this is where the abundance of coarse median structures comes from. These techniques are of independent interest, and we use them in a forthcoming paper to study quotients of short HHGs.