将 Stein 群作为拓扑满群进行研究
Studying Stein's Groups as Topological Full Groups
AI总结：
本文将 Stein 对 Thompson 群 V 的推广纳入拓扑满群框架，统一证明斜率群有限生成时其单导子群有限生成，并计算多类例子的阿贝尔化与有理同调。
摘要：
我们将 Melanie Stein 引入的一类 Thompson 群 $V$ 的推广纳入不断发展的拓扑满群框架中。与 $V$ 一样，Stein 群可描述为具有指定斜率和不可微点的特定分段线性映射。Stein 群包括斜率群由多个整数生成的群类（我们称之为 Stein 整数群），以及斜率群由单个无理数生成的情形，例如 Cleary 群（我们称之为无理斜率 Thompson 群）。我们给出了一个统一的动力学证明：只要斜率群是有限生成的，相关 Stein 群的单导子群就是有限生成的。随后，我们在他人工作的基础上研究 Stein 群的同调。这一研究方向使我们能够具体计算许多例子的阿贝尔化和有理同调。
英文摘要：
We include a class of generalisations of Thompson's group $V$ introduced by Melanie Stein into the growing framework of topological full groups. Like $V$, Stein's groups can be described as certain piecewise linear maps with prescribed slopes and nondifferentiable points. Stein's groups include the class of groups where the group of slopes is generated by multiple integers (which we call Stein's integral groups) and when the group of slopes is generated by a single irrational number, such as Cleary's group, (which we call irrational slope Thompson's groups). We give a unifying dynamical proof that whenever the group of slopes is finitely generated, the simple derived subgroup of the associated Stein's groups is finitely generated. We then study the homology of Stein's groups, building on the work of others. This line of inquiry allows us to compute the abelianisations and rational homology concretely for many examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
