A Nonpositive Curvature Property of Modular Semilattices
arXiv:1905.01449
Abstract
The orthoscheme complex of a graded poset is a metrization of its order complex such that the simplex of each maximal chain is isometric to the Euclidean simplex of vertices . This notion was introduced by Brady and McCammond in geometric group theory, and has applications in discrete optimization and submodularity theory. We address a question of what posets to yield the orthoscheme complex having CAT(0) property. The orthoscheme complex of a modular lattice is shown to be CAT(0), and it is conjectured that this is the case for a modular semilattice. In this paper, we prove this conjecture affirmatively. This result implies that a larger class of weakly modular graphs yields CAT(0) complexes.