Braids, orderings and minimal volume cusped hyperbolic 3-manifolds

TL;DR

This paper explores the relationship between braid group automorphisms, bi-orderability of free groups, and the minimal volume of cusped hyperbolic 3-manifolds, revealing connections between algebraic properties and geometric structures.

Contribution

It demonstrates which braids induce automorphisms preserving bi-orderings and links these to the bi-orderability of fundamental groups of minimal volume hyperbolic 3-manifolds.

Findings

One of the two minimal volume hyperbolic 2-cusped manifolds has a bi-orderable fundamental group.

The other minimal volume 2-cusped manifold does not have a bi-orderable fundamental group.

Pseudo-Anosov braids with minimal dilatation generally do not preserve order.

Abstract

It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas the other does not. We prove a similar result for the 1-cusped case, and have further results for more cusps. In addition, we study pseudo-Anosov braids and find that typically those with minimal dilatation are not order-preserving.