Consider the amalgamated product $A\ast_C B$ of groups such that $A\neq C\neq B$ and both factors $A$, $B$ are finitely generated virtually nilpotent.

Does $A\ast_C B$ always have a subgroup of some finite index $>2$?


  1. There are many results proving residual finiteness (or even subgroup separability) of $A\ast_C B$ under various assumptions. For example, if $C$ is normal in both factors, then $A\ast_C B$ surjects onto $A/C\ast B/C$ which is infinite and residually finite.

  2. There are examples where $A\ast_C B$ is not residually finite, e.g. see A non-Hopfian group by Baumslag.

  3. Baumslag showed that if $A$, $B$ are (finitely generated) torsion free nilpotent, then $A\ast_C B$ surjects onto $\mathbb Z$, and hence has a subgroup of arbitrary index. In fact he proves this for any finitely generated subgroup of such $A\ast_C B$.

The property of having a subgroup of finite index $>2$ is much weaker than residual finiteness, which is why I am hoping for an elementary solution.


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