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The notion of orbifold is quite well established by now. I would like to ask how one should call a point of an orbifold with non-trivial stabilizer? Should one call this a singular point? Of something else?

For some reason, I was not able to find any text that fixes this innocent bit of terminology concerning orbifolds.

Comment. I would like to stress, that I want to know how to call A point (i.e. one point) that has a non-trivial stabilizer. Indeed, as Ryan says in his comment, there is some terminology to define the union of all points with non-trivial stabilizer, but this is not what I am looking for (for example, in algebraic geometry there is a canonical way to call a point that is not smooth, it is called a singular point)

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    $\begingroup$ I think I've seen several different terms. Orbifold locus, or singular strata, for example. $\endgroup$
    – Ryan Budney
    Commented Apr 17, 2012 at 22:21
  • $\begingroup$ You could say 'symmetric point', since it has a non-trivial automorphism group (if the orbifold is represented by a groupoid). But I've not seen this in print, or heard it said. $\endgroup$
    – David Roberts ♦
    Commented Apr 18, 2012 at 2:08
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    $\begingroup$ If you work in algebraic geometry I would strongly advise against singular point, since "singular point" already has a well established meaning on an algebraic variety. Moreover, it suggests misleadingly that the presence of nontrivial isotropy causes the stack to become singular, when morally it is the other way around... the coarse moduli space has no orbifold points, but will in general acquire singularities to compensate. $\endgroup$
    – Dan Petersen
    Commented Apr 18, 2012 at 6:30

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Take a look at Thurston's notes (Chapter 13 of his "The geometry and topology of 3-manifolds", available at the MSRI's site: http://library.msri.org/books/gt3m/). Thurston calls these points singular points. (Same as Satake in his original paper "The Gauss-Bonnet Theorem for V-manifolds" from 1957 where he introduces orbifolds, under the name V-manifolds, see Link) As a geometric topologist, I would use Satake-Thurston's terminology. However, they both work in the geometric topology/Riemannian geometry setting. In the algebro-geometric context you could call them orbifold points to distinguish these points from singular point of the underlying variety.

Google search revealed the following statistics: "stacky point": 328 hits, "orbifold point": 9,750 hits, "singular point" + orbifold: 17,500 hits.

I also like "stucky" and "sticky" points: If you have an isolated singular point $p$ of a Riemannian orbifold, then unparameterized geodesics after entering $p$ get "stuck" there.

On even lighter note, Thurston in his notes explains that the name "orbifold" is the result of a vote he had at his seminar in Princeton. One can have a similar vote on MO to determine what to call points with nontrivial stabilizer.

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  • $\begingroup$ Misha, thanks for your answer. Concerning the vote, indeed, as far as I understand for good of for bad a vote questions would not be accepted on mathoverflow since this would be considered argumentative. $\endgroup$
    – aglearner
    Commented Apr 18, 2012 at 9:37
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"Orbifold point" (probably by analogy with "cone point") is what I hear most frequently, but from your moniker, you are interested in algebraic geometry more than low-dimensional topology, so the nomenclature might be different there.

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I've heard the term "stacky point" said out loud many times. This may well be one of those phrases that's acceptable spoken but nobody wants to be the first to put into print.

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    $\begingroup$ And it is good that it be so! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 18, 2012 at 2:02
  • $\begingroup$ "stacky point" is definitely what I would say. "Definition: A stacky point is a point in the orbifold with nontrivial stabilizer. ..." $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 18, 2012 at 2:30
  • $\begingroup$ Allen, thanks for this answer, I also heard some people using this expression. It sounds quite nice. But I guess I will never use it, at least in this life. Anyway, I would feel bad about using this expression before I learn what is a stack... Is there some place where this notion is explained for dummies (like me)? $\endgroup$
    – aglearner
    Commented Apr 18, 2012 at 22:39
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Fundamental domains of arithmetic Fuchsian groups of the first kind have orbifold structure, and points with non-trivial stabilizers in these cases are called elliptic fixed points. However, I am not sure if this is the terminology used for general orbifolds.

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    $\begingroup$ Johannas, thanks for your answer. Sure, in case of dimension 2 "elliptic point" sounds like a reasonable name. But I never saw this terminology used for orbifolds of dimension more than 2. $\endgroup$
    – aglearner
    Commented Apr 17, 2012 at 22:40
  • $\begingroup$ They are called that way because they are elliptic elements of $\SL_2(\mathbb{R})$. It is usual to call elements in $GL(n)$ or $SL(n)$ with irreducible characteristic polynomial elliptic. $\endgroup$
    – Marc Palm
    Commented Jul 28, 2012 at 19:23
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The question you are asking is a social question, and it cannot have any answer better than a social answer.

Is there a terminology that the orbifold community has settled on and uses universally? No.

Are there common terminologies that are used by various members of the orbifold community? Yes, some of them are listed in other answers.

Given these social realities, how should one refer to a point of an orbifold with non-trivial stabilizer? Pick the one which makes most sense to you for your context. Or, as Misha suggests, pick the one that most honors the historical origins of the concept.

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