On finitely generated left-orderable metabelian groups

A group is left-orderable if it can be equipped with a total order that is invariant under left-multiplication. Such a left-order is computable if there exists an algorithm deciding the order of any given pairs of group elements. The computability problem for left-orders has gained interest in recently years. One of the numerous results is Darbinyan's work on the existence of a solvable group of derived length 3 such that it is left-orderable and has decidable word problem but none of its left-order is computable. On the contrary, Solomon has shown that for free abelian groups, there always exists a computable order. This talk will focus on an ongoing work on the topic of computability of left-orders for metabelian groups.