Computability, orders, and solvable groups

The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length 3. (1) answers a question posed by Downey and Kurtz; (2) answers a question posed by Bludov and Glass in Kourovka Notebook. One of the technical tools used to obtain the main results is a computable version of a subnormal group embedding construction that goes back to Bernhard and Hanna Neumanns and to Philip Hall, and is studied by the author earlier. In this paper we also compliment that result which might be of independent interest.