In what sense is Bass-Serre theory the one-dimensional version of orbifold theory

Question

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.

Answer 1

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 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) in the hyperbolic plane $\mathbb{H}$ by the action of the 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.