TITLE: Subgroups of one-relator groups
QUESTION [10 upvotes]: I know that not every finitely-presented group may be embedded into a one-relator group, for example because of a theorem of Magnus stating that the word problem is solvable in one-relator groups. But does there is a great amount of finitely-generated groups embeddable into a one-relator group?
For instance, is there any (hopefully "large") class of groups known to be embeddable into a one-relator group?

REPLY [3 votes]: It is known that a right-angled Artin group $A(\Gamma)$ embeds in some one-relator group if and only if $\Gamma$ is a forest. In fact, any such $A(\Gamma)$ embeds in the one-relator group $\operatorname{Gp}\langle a,t \mid [a, a^t] = 1\rangle$, which contains a copy of $A(P_4)$ (and hence also the group corresponding to any forest by certain embeddability results). This is all proved in Gray, Robert D., Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups,  ZBL07160162.
Note that in the torsion case, the set of subgroups possible is significantly reduced, and explicit analysis can be done to a greater extent. For example, every $2$-generated subgroup of a $2$-generated one-relator group with torsion is either free or is itself a one-relator group with torsion. This is all possible by the B. B. Newman Spelling Theorem, which provides a significant degree of control. In particular, this theorem implies that one-relator groups with torsion are hyperbolic, so the only right-angled Artin groups embedding in some one-relator group with torsion are free. For detailed combinatorial analysis, and some more examples in the torsion case, see much of the work by Stephen Pride, e.g. Pride, Stephen J., The two-generator subgroups of one-relator groups with torsion, Trans. Am. Math. Soc. 234, 483-496 (1977). ZBL0366.20022.