TITLE: Groups whose finite index subgroups of fixed index are isomorphic
QUESTION [23 upvotes]: I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are isomorphic.
I know three infinite families of such groups: (1) Free abelian groups $\mathbb Z^n$ (2) Free groups $F_n$ (3) Fundamental groups of closed orientable surfaces $\Gamma_n = \pi_1 (\Sigma_n)$.
The way to see that these are indeed examples is via topology: the classifying spaces of these groups are tori, graphs and surfaces and we understand their covering theory.
[Note: some geometric group theorist claim that these are the easiest torsion-free groups that exist. Their outer automorphism groups are given by $GL_n(\mathbb Z)$, $Out(F_n)$ and $Mod^{\pm}_n$, groups that generated a vast body of research.]
Are there more (than the above mentioned) examples of such groups? 
similar, but in fact a different question

REPLY [4 votes]: A recent preprint of Friedl, Park, Petri, Raimbault, and Ray classifies the compact 3-manifolds with empty or toroidal boundary that have the topological analogue of this property. The authors do not know any examples of infinite residually finite groups with the property beyond the three classes listed in the question.