Coxeter Complexes and Graph-Associahedra

TL;DR

This paper introduces a new class of convex polytopes called graph-associahedra, generalizing known polytopes like associahedra and cyclohedra, and explores their geometric and combinatorial properties.

Contribution

It constructs graph-associahedra from arbitrary graphs and demonstrates their role in Coxeter complexes and moduli space compactifications.

Findings

Graph-associahedra generalize associahedra and cyclohedra.

Coxeter complexes can be tiled by graph-associahedra.

The polytopes relate to moduli space compactifications.

Abstract

Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford compactification of the real moduli space of curves.