I'm not very familiar with (even simple examples of) orbifolds, so my first question is:

Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of $C_2$ minus $k$ points ?

My naive answer is: take $\mathbb{C}^*$ minus the same $k$ points. Its fundamental group is freely generated by the $k+1$ loops around the punctures. Now decide that you don't have a "hole" in 0 anymore, but a cone singularity, meaning that the generators corresponding to a loop around 0 is now of order 2. Then I would say that the fundamental group of $C_2$ minus $k$ points is $\langle a_0,\dots,a_k | a_0^2=1\rangle$, ie $Z_2\ltimes F_{2k}$, where $F_{2k}$ is generated by {$a_i,a_0a_ia_0, i\geq 1$}.

Now recall the following construction: take the pure braid group $P_n$ with its standard generators $x_{i,j}, 1\leq i < j\leq n$ given by taking the $j$th strand, letting it go behind all other strand, loop around the $i$th one and going back. Then it's quite easy to see that the subgroup generated by the $x_{i,n}$ is free: it is the subgroup of pure braids for which all but the last strand are fixed straight lines. In fact, it leads to a semidirect product decomposition $P_n=P_{n-1}\ltimes F_{n-1}$. This decomposition is actually a so called "almost direct" product, which is quite an important fact.

This construction has a nice geometric interpretation: let $X_n$ be the configuration space of $n$ points in $\mathbb{C}$, and recall that $P_n=\pi_1(X_n)$. Then the map $X_n \rightarrow X_{n-1}$ which forget the last coordinate is a locally trivial fibration with fiber $\mathbb{C}$ minus $n-1$ points. Then it induces a (split) short exact sequence of fundamental groups

$$1\rightarrow F_{n-1} \rightarrow P_n \rightarrow P_{n-1}\rightarrow 1$$

Let's try to do something similar with the "orbifold braid group" of $C_2$, that is the fundamental group $P_n(C_2)$ of $O_n=${$z_1,\dots,z_n \in C_2, z_i \neq z_j$}.

It seems to me that $P_n(C_2)=P_{n+1}/ \langle x_{1,i}^2=1,i=2 \dots n+1 \rangle$.

The above construction seems to work "at the algebraic level": let $G_n$ be the subgroup of $P_n(C_2)$ generated by (the images of) $x_{i,n+1}$. What is stated in this paper (in a slighty different form) is that $P_n(C_2)=P_{n-1}(C_2) \ltimes G_n$, and that it is an almost direct product too.

But $G_n$ satisfies some relations, for example $x_{i,n+1}$ and $x_{0,n+1}x_{i,n+1}x_{0,n+1}$ commute for a given $i$, hence it is not isomorphic to the fundamental group of $C_2$ minus $n-1$ points (at least if my first naive try is not wrong). While this construction strongly looks like to and shares many algebraic properties with the construction for $P_n$, it does not seems to come from a natural geometric construction. So my real question is:

Am I wrong somewhere ? Is there a natural interpretation of $G_n$ ?

Edit: Here is roughly what happen: assuming that $n=2$ for the sake of simplicity, it doesn't make sense to "freeze" the first strand (and its negative) and to make the second one loop around because the following relation holds:


now pushing the red loop (seen as a loop in the 2-punctured plane) to the bottom plane, we see that it has to be identified with its conjugate by a loop around the two strands at once, ie by the product of the generators of $F_2$. Therefore, this product has to be central, leading to the relation holding in $G_n$ above. So one can ask:

Is there a topological space modelled on this situation, i.e. which looks like to the "complementary in $\mathbb{C}\times[0,1]$ of two strands modulo homotopy". Or at least, is there a way to prove that there are no other relations than declaring that the big loop is central ?

  • $\begingroup$ Hi Adrien, this is not an answer, but have a look at the comment of 11/02/2009 on pages.bangor.ac.uk/~mas010/pstacks.htm $\endgroup$ Commented Jan 10, 2012 at 3:45
  • $\begingroup$ You may want to look at fundamental groups of tautological orbi-bundles whose fibers are orbifolds whose underlying space is the 2-sphere (with several cone point of order 2 and several other points marked) and whose base is the moduli space of such marked orbifolds, where all marked points and cone points have different labels. Such groups will have a normal subgroup of the type you are interested in (free products of cyclic groups of infinite order and order 2) and the quotient group a pure braid group. $\endgroup$
    – Misha
    Commented May 22, 2012 at 14:32
  • $\begingroup$ The link to springerlink.com is broken. I'm also unable to find any snapshot saved on the Wayback Machine. $\endgroup$ Commented Apr 10, 2023 at 0:27
  • 1
    $\begingroup$ @TheAmplitwist It was a while ago, but I'm fairly sure this was "The $D_n$ generalized pure braid group" by Markushevich (link.springer.com/article/10.1007/BF00181653), I'll edit. Beware however, I've been told that their presentation by generators and relation of what is denoted $H_k$ (=$G_n$ in my question), hence their proof, is incorrect although the result itself is correct (see eg arxiv.org/abs/math/0502120). $\endgroup$
    – Adrien
    Commented Apr 11, 2023 at 7:05
  • 1
    $\begingroup$ @TheAmplitwist For a more geometric perspective you might also want to have a look at papers by Roushon, eg arxiv.org/abs/2006.07106 $\endgroup$
    – Adrien
    Commented Apr 11, 2023 at 7:14

1 Answer 1


For the first question, see:

MR0285619 (44 #2837) Hoare, A. Howard M.; Karrass, Abraham; Solitar, Donald. Subgroups of finite index of Fuchsian groups. Math. Z. 120 1971 289–298.

On the first page they talk about what orbifold groups look like (and refer back to Fricke–Klein...)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.