Let $G=A\ast_C B$ be an amalgamated free product of two infinite subgroups over their intersection.

My question:

Is it possible to describe in an 'explicit' way all the subgroups of $G$ of index $2$? Of course, such subgroups are normal.


(1) Recall that subgroups of a free amalgamated product were described by Karrass and Solitar here.

(2) Concerning the subgroup generated by all involutions in the group of automorphisms of the first Weyl algebra (which is the free product of the affine and triangular subgroups), it was shown in Proposition 5b that this group equals the whole group of automorphisms.

(3) This question may be relevant, especially the claim in its answer.

(4) See also this question.

  • $\begingroup$ I don't know whether you can describe them explicitly, but since $[G,G]$ is contained in any such subgroup, the problem reduced to the corresponding problem for abelian groups. $\endgroup$ – Derek Holt May 3 '17 at 8:57
  • $\begingroup$ Thanks for your hint. Can you please elaborate it as an answer? $\endgroup$ – user237522 May 3 '17 at 9:02
  • $\begingroup$ Well it's not exactly an answer, and I don't have much to add at the moment! $\endgroup$ – Derek Holt May 3 '17 at 9:07
  • $\begingroup$ ok, I understand. Thanks again. $\endgroup$ – user237522 May 3 '17 at 9:13
  • $\begingroup$ It is worth noticing that there exist simple groups which decompose as non trivial amalgams of (finitely generated) groups. So it may happen that an amalgam does not contain any subgroup of index two. $\endgroup$ – Seirios May 5 '17 at 16:23

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