# Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the mapping class group $\Gamma_{0,n}$, so that $M_{0,n} \cong T_{0,n} / \Gamma_{0,n}$?

Now $\pi_1(M_{0,n}) \cong \Gamma_{0,n}$ since the Teichmuller space is contractible and the action of $\Gamma_{0,n}$ is free.

What does the Dehn-Nielsen-Baer theorem say in this case? Let $S_{0,n}$ be the $n$ punctured complex projective line then which subgroup of $Out(\pi_1(S_{0,n}))$ does $\Gamma_{0,n}$ correspond to? Also how does $\Gamma_{0,n}$ act on $H_1(S_{0,n})$?

## 1 Answer

Let $S_g$ be a compact Riemann surface of genus $g$ with $n$ marked points $x_1, \ldots, x_n$, and set $S_{g, \, n}:=S_g - \{x_0, \ldots, x_n\}$. Also, denote by $\pi_{g, \, n}$ the fundamental group of $S_{g, \, n}$ (we omit the decoration $n$ when $n=0$).

Then the mapping class group $\Gamma_{g,\, n}$ is defined as $$\Gamma_{g, \, n} := \pi_0 \, \textrm{Diff}^{+}(S_{g, \, n}),$$ where $\textrm{Diff}^{+}(S_{g, \, n})$ is the group of orientation-preserving diffeomorphisms of $S_g$ that fix each $x_i$.

The group $\Gamma_{g, \, n}$ naturally acts on the Teichmueller space $T_{g, \, n}$. Such an action is properly discontinuous and there is a subgroup of finite index acting freely, in such a way that, as an orbifold, $$M_{g, \, n}= T_{g, \, n}/ \Gamma_{g, \, n}.$$ In particular, $\pi^{\textrm orb}_1(M_{g, \, n}) = \Gamma_{g, \, n}$.

By a result of Baer and Nielsen, with this representation $\Gamma_{g}$ is identified with the subgroup of $\textrm{Out}(\pi_{g})$ acting trivially on $H_2(\pi_{g})$. When $n \geq 1$, there are similar characterizations.

For (many) more details, see the paper

R. Hain, E. Looijenga: Mapping class groups and moduli spaces of curves, Algebraic Geometry-Santa Cruz 1995, 97–142, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI (1997),

also available on the arXiv.

• I am exactly interested in the case when $g=0$ and $n \geq 3$. In this case $M_{0,n}$ is actually a manifold and the action of $\Gamma_{0,n}$ is free. In this specific case what would be the subgroup of $Out(\pi_1(S_{0,n})$ – Chitrabhanu Oct 7 '15 at 9:55
• In the case of $n>0$, it's the subgroup that preserves the conjugacy classes that correspond to the boundary components. The book A primer on mapping class groups by Farb--Margalit is the definitive reference. – HJRW Oct 7 '15 at 13:14
• I thought the mapping class in the genus zero case is the pure braid group in n-1 strands? – Venkataramana Oct 7 '15 at 13:31
• $\Gamma_{0, \,n}$ is isomorphic to the mapping class group of the $n$-punctured sphere, see [J. S. Birman, Braid, Links and mapping class groups, Theorem 4.5] – Francesco Polizzi Oct 7 '15 at 13:46