A Nonpositive Curvature Property of Modular Semilattices
Metric Geometry
2019-05-07 v1
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.
Keywords
Cite
@article{arxiv.1905.01449,
title = {A Nonpositive Curvature Property of Modular Semilattices},
author = {Hiroshi Hirai},
journal= {arXiv preprint arXiv:1905.01449},
year = {2019}
}