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Hi,

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:

braid

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 ?

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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 –  yael fregier Jan 10 '12 at 3:45
    
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. –  Misha May 22 '12 at 14:32

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...)

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