Abstract: A left-order on a group is a linear order which is invariant under the left translations, and a bi-order is a linear order which is invariant under both left and right translations. Having a torsion element in a group is an obstruction to left-orderability; similarly, having a generalized torsion is an obstruction to bi-orderability. For many classes of groups (e.g. one-relator groups), the absence of torsion implies left-orderability. Similarly, the absence of generalized torsion often implies bi-orderability.
In the first part of the talk, I'll briefly discuss the use of orders in topology (primarily, in 3-dimensional topology). In the second part, I will present an example of a non-bi-orderable one relator group without a generalized torsion. This is a joint work with James Thorne.
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