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首页> 外文期刊>Journal of Algebraic Combinatorics >On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups
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On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

机译：关于非阿贝尔有限简单群的迭代花圈乘积的逆极限的最小拓扑生成集的一些通用构造

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摘要

Let ((H_i)_{ige 1}) be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit (ldots wr H_2wr H_1) of iterated permutational wreath products of the groups (H_i). The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group (H) on a set (X). We show that the wreath power (ldots wr Hwr H) contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet (X). The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup.
机译：令（（H_i）_ {ige 1}）是非阿贝尔有限简单传递性置换群的任意序列。通过使用时变自动机的组合语言，我们提供了两个元素集的显式且自然定义的构造，该元素集在组（H_i）的迭代置换花圈乘积的反极限（ldots wr H_2wr H_1）中生成密集子组。 ）。相应的自动机配备了三个状态，其中一个是中性的，其他两个状态生成的半群是自由的。我们推导了由该自动机生成的群的其他代数和几何性质。通过使用Mealy自动机的概念，我们获得了集合（X）上任意非阿贝尔有限简单传递性置换群（H）的无限置换花圈能力的类似构造。我们表明，花环幂（ldots wr Hwr H）包含一个密集的2生成的不成比例表示的指数增长的子集，该子集是由字母（X）上的3状态Mealy自动机生成的。该自动机产生的自相似基团是自我复制，收缩的并且在换向器子组上有规律地弱分支。

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