Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds

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 $F_n$ of 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 groupwhereas 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.