We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective and strongly shortcut spaces. show that every space admits a new is injective. The quasi-isometric to the original preserved under automorphisms space. uniformly bounded geometry shortcut. Consequently, groups -- including mapping class surfaces are Using these results, we deduce ...

Establishing that a demand mapping is injective core first step for variety of methodologies. When version the law holds, global injectivity can be checked by seeing whether constant over any line segments. we add assumption differentiability, obtain necessary and sufficient conditions generalize classical \cite{gale1965jacobian} quasi-definite Jacobians.

Abstract We study distinguished objects in the category $\mathcal{B}\mathcal{L}$ of Banach lattices and lattice homomorphisms. The free construction introduced by de Pagter Wickstead [ 8] generates push-outs, combining this with an old result Kellerer 17] on marginal measures, amalgamation property is established. This will be key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ separably $\mathc...