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?
There have been a couple of recent results which demonstrate that the class of subgroups of one-relator groups is very rich. For instance, Calegari–Walker proved that a random 1-relator group contains a surface subgroup, and Calegari and I improved this by demonstrating that a random 1-relator group contains a subgroup isomorphic to the fundamental group of an acylindrical hyperbolic 3-manifold. Both papers are on the arXiv.
A random 1-relator group is hyperbolic (since its relator satisfies the C'(1/6) small-cancellation condition) and 2-dimensional. Results of Bowditch and Kapovich–Kleiner then impose certain restrictions on the (quasiconvex) subgroups that can arise—see the answer to Topology of boundaries of hyperbolic groups for some details. In a sense, the above results show that the quasiconvex subgroups of random 1-relator groups are as rich as possible. This is discussed in the introduction to our paper.
This leaves two strands uncovered. A random 1-relator group will also contain non-quasiconvex finitely generated subgroups. (I think Dunfield–Thurston proved this in the 2-generator case.) And, of course, plenty of 1-relator groups do not behave like a ‘random’ one—for instance, Baumslag's famous example of a non-abelian 1-relator group with every finite quotient cyclic.
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.
