The Polyhedral Tree Complex

TL;DR

This paper links the tree complex with convex polytopes like associahedra and cyclohedra, showing the tree complex as a barycentric subdivision of a polyhedral complex built from these polytopes, with applications to mapping class groups.

Contribution

It provides a novel characterization of associahedra and cyclohedra via planar tree embeddings and relates the tree complex to a new polyhedral cell complex structure.

Findings

Tree complex is the barycentric subdivision of a polyhedral complex.

Cells of the complex are products of associahedra and cyclohedra.

Connection established between tree complex and convex polytopes.

Abstract

The tree complex is a simplicial complex defined in recent work of Belk, Lanier, Margalit, and Winarski with natural applications to mapping class groups and complex dynamics. In this article, we connect this setting with the study of certain convex polytopes: associahedra and cyclohedra. Specifically, we describe a characterization of these polytopes using planar embeddings of trees and show that the tree complex is the barycentric subdivision of a polyhedral cell complex for which the cells are products of associahedra and cyclohedra.