Computability, orders, and solvable groups

TL;DR

This paper constructs specific bi-orderable and solvable groups demonstrating undecidability and non-computability properties, answering open questions in group theory and computability.

Contribution

It introduces the first examples of computable bi-orderable groups without computable bi-orderings and solvable groups with undecidable word problems, solving longstanding open problems.

Findings

Existence of a computable bi-orderable group without a computable bi-ordering

Existence of a solvable, two-generated group with an undecidable word problem

Both groups are among two-generated solvable groups of derived length 3

Abstract

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 computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest.