On the Involution Generators of the Mapping Class Group of a Punctured Surface

Let $\Sigma_{g,p}$ denote a connected orientable surface of genus-$g$ with $p$-punctures, when $p=0$ we write $\Sigma_{g}$. The mapping class group of $\Sigma_{g,p}$ is the group of isotopy classes of orientation-preserving homeomorphisms of $\Sigma_{g,p}$ preserving the set of punctures.

Here is a brief history of generating sets for ${\rm Mod}(\Sigma_{g,p})$: Dehn [4] showed that ${\rm Mod}(\Sigma_{g})$ can be generated by $2g(g-1)$ Dehn twists. About a quarter century later, Lickorish [15] gave a generating set consisting of $3g-1$ Dehn twists. Later, Humphries [8] reduced the number of Dehn twist generators to $2g+1$. He also proved that the number $2g+1$ is minimal for $g\geq 2$. Johnson [10] proved that the same set of Dehn twists also generates ${\rm Mod}(\Sigma_{g,1})$. In the presence of multiple punctures, Gervais [7] proved that ${\rm Mod}(\Sigma_{g,p})$ can be generated by $2g+p$ Dehn twists for $p\geq 1$.

If it is not required that the generators are Dehn twists, then it is possible to obtain smaller generating sets for ${\rm Mod}(\Sigma_{g,p})$: For $g\geq 3$ and $p=0$, Lu [17, Theorem $1.3$] proved that ${\rm Mod}(\Sigma_{g})$ can be generated by three elements.For $g\geq 1$ and $p=0$ or $1$, a minimal (since the group is not abelian) generating set of two elements, a product of two Dehn twists and a product of $2g$ Dehn twists, was first given by Wajnryb [25]. Korkmaz [12, Theorem $5$] improved this result by showing that one of these two generators can be taken as a Dehn twist. He also showed that this group is generated by two elements of finite order  [12, Theorem $14$]. For $g\geq 3$, Kassabov obtained a generating set of involution elements where the number of generators depends on $g$ and the parity of $p$ (see  [11, Theorem 1]). Later, Monden [20] removed the parity condition on $p$ for $g=7$ and $g=5$. For $g\geq 1$ and $p\geq 2$, Monden [21] also gave a generating set for ${\rm Mod}(\Sigma_{g,p})$ consisting of three elements. Recently, he [22] gave a minimal generating set for ${\rm Mod}(\Sigma_{g,p})$ containing two elements for $g\geq 3$.

Note that any infinite group generated by two involutions must be isomorphic to the infinite dihedral group whose subgroups are either cyclic or dihedral. Since ${\rm Mod}(\Sigma_{g,p})$ contains nonabelian free groups, it cannot be generated by two involutions. In this paper, we obtain the following result (cf.  [11, Remark 5]):

At the end of the paper, we also show that Theorem A also holds for the cases $p=2$ or $p=3$. For surfaces with odd number of punctures, we have the following result:

The paper is organized as follows: In Section 2, we quickly provide the necessary background on mapping class groups. The proofs of Theorem A and Theorem B are given in Section 3.

Let $\Sigma_{g,p}$ denote a connected orientable surface of genus $g$ with $p$ punctures specified by the set $P=\{z_{1},z_{2},\ldots,z_{p}\}$ of $p$ distinguished points. If $p$ is zero then we omit it from the notation and write $\Sigma_{g}$. The mapping class group ${\rm Mod}(\Sigma_{g,p})$ of the surface $\Sigma_{g,p}$ is defined to be the group of the isotopy classes of orientation preserving self-diffeomorphisms of $\Sigma_{g,p}$ which fix the set $P$. The mapping class group ${\rm Mod}(\Sigma_{g,p})$ of the surface $\Sigma_{g,p}$ is defined to be the group of isotopy classes of all orientation preserving self-diffeomorphisms of $\Sigma_{g,p}$ which fix the set $P$. Let ${\rm Mod}_{0}(\Sigma_{g,p})$ denote the subgroup of ${\rm Mod}(\Sigma_{g,p})$ consisting of elements which fix the set $P$ pointwise. It is obviuos that we have the following exact sequence:

where $Sym_{p}$ denotes the symmetric group on the set $\{1,2,\ldots,p\}$ and the last projection is given by the restriction of the isotopy class of a diffeomorphism to its action on the punctures.

Let $\beta_{i,j}$ be an embedded arc that joins two punctures $z_{i}$ and $z_{j}$ and does not intersect $\delta$ on $\Sigma_{g,p}$. Let $D_{i,j}$ denote a closed regular neighbourhood of $\beta_{i,j}$, which is a disk with two punctures. There exists a diffeomorphism $H_{i,j}:D_{i,j}\to D_{i,j}$, which interchanges the punctures such that $H_{i,j}^{2}$ is equal to the right handed Dehn twist about $\partial D_{i,j}$ and is the identity on the complement of the interior of $D_{i,j}$. Such a diffeomorphism is said to be the (right handed) half twist about $\beta_{i,j}$. It can be extended to a diffeomorphism of ${\rm Mod}(\Sigma_{g,p})$. Throughout the paper we do not distinguish a diffeomorphism from its isotopy class. For the composition of two diffeomorphisms, we use the functional notation; if $f$ and $g$ are two diffeomorphisms, then the composition $fg$ means that $g$ is applied first and then $f$.
For a simple closed curve $a$ on $\Sigma_{g,p}$, following  [1, 13], we denote the right-handed Dehn twist $t_{a}$ about $a$ by the corresponding capital letter $A$. Let us also remind the following basic facts of Dehn twists that we use frequently throughout the paper: Let $a$ and $b$ be simple closed curves on $\Sigma_{g,p}$ and $f\in{\rm Mod}(\Sigma_{g,p})$.

• •

  If $a$ and $b$ are disjoint, then $AB=BA$ (Commutativity).

• •

  If $f(a)=b$, then $fAf^{-1}=B$ (Conjugation).

Let us finish this section by noting that we denote the conjugation relation $fgf^{-1}$ by $f^{g}$ for any $f,g\in{\rm Mod}(\Sigma_{g,p})$.

Let us start this section by recalling the following set of generators given by Korkmaz [13, Theorem $5$].

Let us also recall the following well known result from algebra.

In our case where $G={\rm Mod}(\Sigma_{g,p})$ and $N={\rm Mod}_{0}(\Sigma_{g,p})$, we have the following short exact sequence:

Therefore, we obtain the following useful result which follows immediately from Lemma 3.2. Let $\Gamma$ be a subgroup of ${\rm Mod}(\Sigma_{g,p})$. If the subgroup $\Gamma$ contains ${\rm Mod}_{0}(\Sigma_{g,p})$ and has a surjection to $Sym_{p}$ then $\Gamma={\rm Mod}(\Sigma_{g,p})$.

Throughout the paper, we consider the embeddings of $\Sigma_{g,p}$ into $\mathbb{R}^{3}$ in such a way that it is invariant under the rotations $\rho_{1}$ and $\rho_{2}$. Here, $\rho_{1}$ and $\rho_{2}$ are the rotations by $\pi$ about the $z$-axis (see Figures 1 and 2). Note that ${\rm Mod}(\Sigma_{g,p})$ contains the element $R=\rho_{1}\rho_{2}$ which satisfies the following:

• (i)

  $R(a_{i})=a_{i+1}$, $R(b_{i})=b_{i+1}$ for $i=1,\ldots,g-1$ and $R(b_{g})=b_{1}$,

• (ii)

  $R(c_{i})=c_{i+1}$ for $i=1,\ldots,g-2$,

• (iii)

  $R(z_{1})=z_{p}$ and $R(z_{i})=z_{i-1}$ for $i=2,\ldots,p$.

We want to note here that, in the following lemmata, where we present generating sets for surfaces with even number of punctures, we mainly follow the proof of [13, Theorem $5$]. We use them in the proof of Theorem A and then for surfaces with odd number of punctures we explain how our arguments are modified.

If $g$ is odd and $p$ is even, we have the following result:

In the following four lemmata, we give generating sets for smaller genera.