Suppose that $G, H$ are **finitely generated** groups such that $H$ is isomorphic to a finite index subgroup of $G$ and vice versa. Does it follow that $G$ is isomorphic to $H$?

I am sure that the answer is negative but cannot find an example. I am mostly interested in the case of finitely presented groups. The assumption of finite index is, of course, necessary, otherwise one can take any two nonabelian free groups of finite rank. Here is what I know: Given a pair of groups $G, H$ as above, there is, of course, a sequence of isomorphic proper subgroups of finite index

$$
... G_n\lneq G_{n-1}\lneq ...\lneq G_1\lneq G
$$
One can rule out the existence of such a sequence when $G$ is nonelementary hyperbolic, but this does not say much.

**PS.** Noam's example makes me feel rather silly since I have seen such groups *in vivo*: The affine Coxeter groups $\tilde{B}_n, \tilde{C}_n$, $n\ge 3$.

Graphe de groupes et groupes co-hopfiens. Removing the assumption of being one-ended is done inQuasiregular self-mappings of manifolds and word hyperbolic groupsbut only in the torsion-free case. I didn't read Lemma 4.2 carefully yet, but it seems reasonable to think that a similar argument can be followed for groups splitting over finite subgroups.The articleComplexity volumes of splittable groupsmay also be relevant. $\endgroup$5more comments