生成简单活跃群
Generating simple vigorous groups
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中文总结 AI 辅助
研究简单活跃群的生成问题，通过构造性证明得出该类群的多个强生成结果，如由三个对合生成等，证明汤普森群V是(2,3)生成的，还强化了相关嵌入定理，建立了新的生成准则。
中文摘要 AI 辅助
简单活跃群构成了康托空间同胚群的一个广泛类别，包括汤普森群V及其各种推广形式以及许多其他群，如涅克拉舍维奇的动力起源群。Bleak、Elliott和Hyde（2024）证明了每个有限生成的简单活跃群都是2生成的。本文给出了这类简单群的几个强生成结果，例如由三个对合生成、由任意m≥2和n≥3的阶为m和阶为n的元素生成等。这些结果类似于有限单群的著名结果但证明不同。还得出汤普森群V是(2,3)生成的等结论，证明具有建设性并建立了新的生成准则。
英文摘要
The simple vigorous groups form a broad class of groups of homeomorphisms of Cantor space that includes Thompson's group $V$, its various generalisations and many others such as Nekrashevych's groups of dynamical origin. Bleak, Elliott and Hyde (2024) proved that every finitely generated simple vigorous group is $2$-generated, and, in this paper, we give several strong generation results for this class of simple groups. For example, we prove that if $G$ is a finitely generated simple vigorous group, then $G$ is generated by three involutions, $G$ is generated by an element of order $m$ and an element of order $n$ for any choice of $m \geq 2$ and $n \geq 3$, $G$ has a minimal generating set of size $k$ for all $k \geq 2$, every nontrivial element of $G$ is contained in a generating pair and the direct power $G^n$ is $2$-generated for all $n$. These results are analogous to well-known results for finite simple groups, but of course the proofs in this context are quite different. One consequence of our results is that Thompson's group $V$ is $(2, 3)$-generated, which answers a question of Sapir (2017). Another consequence is that every finitely generated group quasi-isometrically embeds in a $(2, 3)$-generated simple group, strengthening theorems of Hall (1974) and Bridson (1998). All of our proofs are constructive, and we establish several new generation criteria for these groups, which we expect to be of wider interest, even just for Thompson's group $V$.
