The Wikipedia article on Bass-Serre theory claims that graphs of groups (in the context of Bass-Serre theory) "can be viewed as one dimensional versions of orbifolds." I hazily see a connection between a graph of groups and the notion of an orbifold, but I have no concrete sense of what the connection is nor how it is a 1 dimensional analogue. This is my first post on mathoverflow as a graduate student so apologies if my question is ill-posed or too basic.



In what sense is Bass-Serre theory the one-dimensional version of orbifold theory
A graph is a $1$-dimensional manifold with singularities and a graph of groups is, roughly, a $1$-dimensional orbifold with singularities. Every graph of groups has a [Bass-Serre covering tree](https://en.wikipedia.org/wiki/Bass%E2%80%93Serre_theory#Bass%E2%80%93Serre_covering_trees) which is its universal cover as an orbifold, and it can be written as the quotient of this tree by its fundamental group $\pi\_1$. This is exactly analogous to presenting an orbifold as a quotient of a manifold by a group action (when such a presentation exists).


Essentially the only example I understand in any detail is the graph of groups given by a copy of $C\_2$ attached to a copy of $C\_3$ by an edge, which, thought of as an orbifold, is the quotient of a certain tree (which I think is pictured [here](https://en.wikipedia.org/wiki/Modular_group#Tessellation_of_the_hyperbolic_plane)) in the hyperbolic plane $\mathbb{H}$ by the action of the [modular group](https://en.wikipedia.org/wiki/Modular_group) $C\_2 \ast C\_3 \cong PSL\_2(\mathbb{Z})$. You can see some pictures of this orbifold and some of its finite covers (corresponding to finite index subgroups of the modular group) in [this blog post](https://qchu.wordpress.com/2015/11/29/drawing-subgroups-of-the-modular-group/).


