Continuous Noncrossing Partitions and Weighted Circular Factorizations

TL;DR

This paper introduces continuous noncrossing partitions on the unit circle, linking them to weighted linear factorizations and topological structures related to braid and Artin groups, extending to Coxeter groups.

Contribution

It defines a new class of continuous noncrossing partitions, establishes their topological properties, and connects them to dual Garside and Artin group structures in a broad Coxeter group context.

Findings

Continuous noncrossing partitions form a topological poset.

Maximal elements form a space homeomorphic to dual Garside classifying space.

Generalization to Coxeter groups yields spaces related to dual Artin groups.

Abstract

This article examines noncrossing partitions of the unit circle in the complex plane; we call these continuous noncrossing partitions. More precisely, we focus on the degree-$d$ continuous noncrossing partitions where unit complex numbers in the same block have identical $d$-th powers. We prove that the degree-$d$ continuous noncrossing partitions form a topological poset whose uncountable set of elements can be indexed by equivalence classes of objects we call weighted linear factorizations of factors of a $d$-cycle. Moreover, the maximal elements in this poset form a subspace homeomorphic to the dual Garside classifying space for the $d$-strand braid group. The degree-$d$ continuous noncrossing partitions of the unit circle are a special case of a more general construction. For every choice of Coxeter element $c$ in any Coxeter group $W$ we define a topological poset of equivalence classes of weighted linear factorizations of factors of $c$ in $W$ whose elements we call continuous $c$-noncrossing partitions. The maximal elements in this poset form a subspace homeomorphic to the one-vertex complex whose fundamental group is the corresponding dual Artin group.