Geodesic
Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds
Eiko Kin1 ; Dale Rolfsen2
1Osaka University, Toyonaka, Japan
2Pacific Institute for the Mathematical Sciences and University of British Columbia, Vancouver, Canada
Groups, geometry, and dynamics, Tome 12 (2018) no. 3, pp. 961-1004

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DOI

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 Fn​ 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.
DOI : 10.4171/ggd/460
Classification : 20-XX, 57-XX
Mots-clés : Ordered groups, bi-orderings, knots and links, braid groups, free groups, mapping class groups, fibered 3-manifolds

Eiko Kin  1   ; Dale Rolfsen  2

1 Osaka University, Toyonaka, Japan
2 Pacific Institute for the Mathematical Sciences and University of British Columbia, Vancouver, Canada 
Eiko Kin; Dale Rolfsen. Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds. Groups, geometry, and dynamics, Tome 12 (2018) no. 3, pp. 961-1004. doi: 10.4171/ggd/460
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     title = {Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds},
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