Let $G$ and $H$ be infinite subgroups of some ambient group, and let $\Gamma$ denote their intersection. If $\[\Gamma :G\]$ and $\[\Gamma: H\]$ are finite, so that $G$ and $H$ are commensurable, when does it hold that $\gcd(\[\Gamma :G\],\[\Gamma: H\])=1$? Does anybody have any ideas about how one might approach a problem like this?
This is always the case if $G$ (thus also $H$) is an infinite cyclic group because $\Gamma$ is generated by $g^N=h^M$ for some pair of co-prime integers $M,N$. If this is a silly question please excuse me- I am not a group theorist and I don't even have any intuition for how hard this question is!