AI 中文总结
研究映射类群中两个映射类生成同构于鲍姆施拉格 - 索利塔群的子群的条件，给出充分必要条件，构造特定同构子群，并证明相关子群满足广义尼尔森实现。
AI 中文摘要
对于$g\geq2$以及非零整数$p,q$，设$\mathrm{Mod}(S_g)$为亏格为$g$的闭可定向曲面$S_g$的映射类群，$\mathrm{BS}(p,q)$为鲍姆施拉格 - 索利塔群。我们给出了两个映射类$G,F\in\mathrm{Mod}(S_g)$生成同构于$\mathrm{BS}(p,q)$的子群的充分必要条件。特别地，若$\mathrm{BS}(p,q)$嵌入$\mathrm{Mod}(S_g)$，则$|p| = |q|$且$G,F$是无限阶的可约映射类。我们还构造了同构于$\mathrm{BS}(p,p)$和$\mathrm{BS}(p, - p)$（$p>1$）的子群。最后表明$\mathrm{Mod}(S_g)$的每个无限亚循环子群和鲍姆施拉格 - 索利塔子群都能提升为$\mathrm{Diff}^+(S_g)$的同构子群，即这些子群满足广义尼尔森实现。
英文摘要
For $g\geq 2$ and nonzero integers $p,q$, let $\mathrm{Mod}(S_g)$ be the mapping class group of a closed oriented surface $S_g$ of genus $g$, and let $\mathrm{BS}(p,q)$ be the Baumslag-Solitar group. We provide necessary and sufficient conditions under which two mapping classes $G,F\in \mathrm{Mod}(S_g)$ generate a subgroup isomorphic to $\mathrm{BS}(p,q)$. In particular, if $\mathrm{BS}(p,q)$ embeds in $\mathrm{Mod}(S_g)$, then $|p|=|q|$ and $G,F$ are reducible mapping classes of infinite order. We also construct subgroups isomorphic to $\mathrm{BS}(p,p)$ and $\mathrm{BS}(p,-p)$ for $p>1$. Finally, we show that every infinite metacyclic subgroup and every Baumslag-Solitar subgroup of $\mathrm{Mod}(S_g)$ lifts to an isomorphic subgroup of $\mathrm{Diff}^+(S_g)$, that is, these subgroups satisfy the generalized Nielsen realization.
Comments7 pages, 4 figures
