Model Theory and Algorithms for One-Relator Groups

We study the elementary theories and algorithmic properties of one-relator groups. We give a first-order sentence characterizing Baumslag-Solitar-freeness in this class. For fixed $|m|>1$, we regularly interpret \BS(1,m) as a definable subgroup in every one-relator group containing it. We uniformly, regularly and injectively interpret integer arithmetic in countable groups of cohomological dimension at most two. We obtain one uniform interpretation for all one-relator groups containing a non-unimodular Baumslag–Solitar subgroup. Every fixed finitely presented group is regularly interpretable in these groups, and their elementary theories are undecidable. We also describe the GBS overgroups of a non-unimodular Baumslag–Solitar subgroup. For the unimodular groups we construct finite-index direct products of free groups and $\Z$, and reduce the elementary theory of \BS(k,k) to that of a free group with a named periodic automorphism. Using the recent Open AI result that BS-free one-relator groups are hyperbolic we obtain a terminating algorithm recognizing this class and obtain consequences. From the virtual compact specialness result of another Open AI paper, we obtain an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel may have infinite rank. For classes of torsion-free one-relator groups in which CSA implies toral relative hyperbolicity, we give a uniform algorithm deciding CSA and, in the affirmative case, the existence of a subgroup isomorphic to $\Z^2$.