相关论文: On the involution generators of the mapping class …

We prove that the extended mapping class group, $\rm Mod^{*}(\Sigma_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\rm…

Let $\textrm{Mod}(N_{g, p})$ denote the mapping class group of a nonorientable surface of genus $g$ with $p$ punctures. For $g\geq14$, we show that $\textrm{Mod}(N_{g, p})$ can be generated by five elements or by six involutions.

Let $\Sigma_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(\Sigma_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm…

Let $\Sigma_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ is generated by involutions. He also…

Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many…

Let $\Sigma_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(\Sigma_{\textit{g,p}})$ be the mapping class group of $\Sigma_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm…

We prove that the mapping class group of a closed connected orientable surface of genus $g$ is generated by three involutions for $g\geq 6$.

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$,…

We prove that, for $g\geq19$ the mapping class group of a nonorientable surface of genus $g$, $\textrm{Mod}(N_g)$, can be generated by two elements, one of which is of order $g$. We also prove that for $g\geq26$, $\textrm{Mod}(N_g)$ can be…

Let $\Sigma_{g,p}$ be a oriented connected surface of genus $g$ with $p$ punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the mapping class group and the extended mapping class group of $\Sigma_{g,p}$, respectively.…

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate…

We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.

Let S = S(n) denote the infinite surface with n ends, n \in N, accumulated by genus. For n \geq 6, we show that the mapping class group of S is topologically generated by five involutions. When n \geq 3, it is topologically generated by six…

Let $N_g$ be a closed, connected, nonorientable surface of genus $g$. We prove that for $g \ge 13$, the mapping class group $\text{Mod}(N_g)$ can be generated by exactly two elements. This improves the previously known bound of $g \ge 19$.

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…

Let $N_{g}$ denote the closed non-orientable surface of genus $g$ and let ${\mathcal M} _g$ denote the mapping class group of $N_{g}$. Let ${\mathcal T} _g$ denote the twist subgroup of ${\mathcal M} _g$ which is the subgroup of ${\mathcal…

We study torsion generators for the (extended) mapping class group or the extended mapping class group of a closed connected orientable surface of genus g. We show that for every g is grater than or equal to 14, mapping class group can be…

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements.…

We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.

This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup,…