Computability, Orders, and Solvable Groups

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 computably presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$.(1) [a]nswers 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.

Regular left-orders on groups

“…First, not all left-orderable groups admit RE-left-orders. This has been recently proved by Harrison-Trainor [14] for left-ordererable groups and, in the bi-orderable case by Darbinyan [10]. Moreover, the lack of RE-left-orders is not related to the solvability of the word problem.…”

“…The result about regularity of left-orders on polycyclic groups cannot be promoted to the case of solvable groups [10]. Here we give the complete picture for when a solvable Baumslag-Solitar groups BS(1, q), q ∈ Z−{0} admits a regular positive cone.…”

“…This paper discusses the computational complexity of left-orders (or equivalently positive cones). This is a topic that has gained interest in the recent years: Darbiyan [10] and Harrison-Trainor [14] have constructed groups with solvable word problems but no computable left-orders. Rourke and Weist have studied the complexity of left-orders on certain mapping class groups [28].…”

Embeddings into left‐orderable simple groups

“…By [19], there is a left‐orderable computable group without any computable left‐order. In fact, there is a finitely generated orderable computable group without any computable order [9]. Example The natural order on the group of rational numbers is computable.…”

“…Also, the existence of finitely generated left‐orderable groups with decidable word problem but without recursively enumerable positive cone is first shown in [ 9 ]. Earlier, the analogous result for countable but not finitely generated groups was shown in [ 19 ].…”