Remark on ordered braid groups

TL;DR

This paper links the algebraic orderings of braid groups to geometric structures on surfaces, using cluster C*-algebras and Teichmüller space, revealing deep connections between algebra, geometry, and dynamics.

Contribution

It introduces a novel connection between the Dehornoy order on braid groups and cluster C*-algebras associated to surfaces, and characterizes the space of left-orderings of surface fundamental groups within Teichmüller space.

Findings

Dehornoy order recovered from cluster C*-algebra tracial states

Space of left-orderings is a dense, totally disconnected subset of projective Teichmüller space

Each left-ordering corresponds to a Riemann surface orbit under geodesic flow

Abstract

We recover the Dehornoy order on the braid group $B_{2g+n}$ from the tracial state on a cluster $C^*$-algebra $\mathbb{A}(S_{g,n})$ associated to the surface $S_{g,n}$ of genus $g$ with $n$ boundary components. It is proved that the space of left-ordering of the fundamental group $\pi_1(S_{g,n})$ is a totally disconnected dense subspace of the projective Teichm\"uller space $\mathbb{P}T_{g,n}\cong \mathbf{R}^{6g-7+2n}$. In particular, each left-ordering of $\pi_1(S_{g,n})$ defines the orbit of a Riemann surface $S_{g,n}$ under the geodesic flow on the space $T_{g,n}$.