Branch actions and the structure lattice

Jorge Fariña-Asategui, Rostislav Grigorchuk
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Abstract

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.
分支行动和结构网格
J.威尔逊(J. S. Wilson)在 1971 年证明了属于他的第二类群的结构晶格与康托尔三元组的开子集布尔代数之间的同构关系。在本文中,我们将这种同构关系推广到分支群类。此外,我们还证明,对于一个群 $G$ 在球面同根树 $T$ 上的每一个忠实分支作用,在与 $G$ 的结构晶格相关的布尔代数和 $T$ 边界的闭合子集的布尔代数之间存在一个规范的 $G$-等价同构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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