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 -strand braids give rise to automorphisms which preserve some bi-ordering of the free group of rank . 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.
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Eiko Kin, Dale Rolfsen, Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds. Groups Geom. Dyn. 12 (2018), no. 3, pp. 961–1004DOI 10.4171/GGD/460