7
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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 this MO question 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 no-abelian 1-relator group with every finite quotient cyclic.

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ A couple more comments. These results are not embeddability results as asked for in the final paragraph of the question, in the sense that they don't say 'Every group of class X is embeddable into a 1-relator group', but rather 'A random 1-relator group contains a subgroup from class X'. But the proof techniques do generate a very large number of subgroups of this form---a generic 1-relator group contains many 3-manifold groups as subgroups. $\endgroup$ – HJRW Oct 4 '14 at 18:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.