Non-standard bi-orders on punctured torus bundles

Results of Perron and Rolfsen imply that untwisted hyperbolic once-punctured torus bundles over the circle have bi-orderable fundamental groups. They do this by showing that the action of the monodromy preserves a “standard” bi-ordering formed using the lower central series of the free group. Here we investigate other bi-orderings that punctured torus bundle groups can have. We show that for every such bi-ordering, the largest and second largest proper convex subgroups match the corresponding convex subgroups for a standard bi-ordering. Moreover, if there exists a third largest convex subgroup, it must also match the third largest convex subgroup for a standard bi-ordering. However, we also show that these groups admit non-standard bi-orderings.