# A Nonpositive Curvature Property of Modular Semilattices

## TL;DR

This paper proves that the orthoscheme complex of a modular semilattice has the CAT(0) property, extending known results from modular lattices and impacting the study of weakly modular graphs.

## Contribution

It confirms that the orthoscheme complex of a modular semilattice is CAT(0), a conjecture previously unproven, broadening the class of posets with this property.

## Key findings

- Orthoscheme complex of a modular lattice is CAT(0)
- Conjecture that modular semilattices have CAT(0) orthoscheme complexes is proven
- Implication for weakly modular graphs and their complexes

## 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 $0, e_1,e_1+e_2,\ldots, e_1+e_2+ \cdots + e_n$. 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.